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Theorem mulsasslem2 28550
Description: Lemma for mulsass 28552. Expand the right hand side of the formula. (Contributed by Scott Fenton, 9-Mar-2025.)
Hypotheses
Ref Expression
mulsasslem.1 (𝜑 → 𝐴 ∈ No )
mulsasslem.2 (𝜑 → 𝐵 ∈ No )
mulsasslem.3 (𝜑 → 𝐶 ∈ No )
Assertion
Ref Expression
mulsasslem2 (𝜑 → (𝐴 ·s (𝐵 ·s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}))))
Distinct variable groups:   𝐴,𝑎,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅,𝑧𝐿,𝑧𝑅   𝐵,𝑎,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅,𝑧𝐿,𝑧𝑅   𝐶,𝑎,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅,𝑧𝐿,𝑧𝑅
Allowed substitution hints:   𝜑(𝑎, 𝑥𝐿, 𝑥𝑅, 𝑦𝐿, 𝑦𝑅, 𝑧𝐿, 𝑧𝑅)
Proof of Theorem mulsasslem2
Dummy variables 𝑏 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lltr 28248 . . . 4 ( L ‘𝐴) <<s ( R ‘𝐴)
21a1i 11 . . 3 (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
3 mulsasslem.2 . . . 4 (𝜑 → 𝐵 ∈ No )
4 mulsasslem.3 . . . 4 (𝜑 → 𝐶 ∈ No )
53, 4mulcut2 28519 . . 3 (𝜑 → ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}) <<s ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))}))
6 mulsasslem.1 . . . . 5 (𝜑 → 𝐴 ∈ No )
7 lrcut 28290 . . . . 5 (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
86, 7syl 18 . . . 4 (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
98eqcomd 2767 . . 3 (𝜑 → 𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
10 mulsval2 28497 . . . 4 ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 ·s 𝐶) = (({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}) |s ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})))
113, 4, 10syl2anc 596 . . 3 (𝜑 → (𝐵 ·s 𝐶) = (({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}) |s ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})))
122, 5, 9, 11mulsunif 28536 . 2 (𝜑 → (𝐴 ·s (𝐵 ·s 𝐶)) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))}) |s ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))})))
13 unab 4254 . . . . 5 ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))}
14 r19.43 3131 . . . . . . 7 (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))))
15 rexun 4142 . . . . . . . . 9 (∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
16 eqeq1 2765 . . . . . . . . . . . . 13 (𝑏 = 𝑡 → (𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ↔ 𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))
17162rexbidv 3228 . . . . . . . . . . . 12 (𝑏 = 𝑡 → (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))
1817rexab 3653 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ ∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
19 rexcom4 3290 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
20 rexcom4 3290 . . . . . . . . . . . . . 14 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
21 ovex 7453 . . . . . . . . . . . . . . . 16 (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∈ V
22 oveq2 7428 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) → (𝐴 ·s 𝑡) = (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))
2322oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) → ((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) = ((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
24 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) → (𝑥𝐿 ·s 𝑡) = (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))
2523, 24oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) → (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
2625eqeq2d 2772 . . . . . . . . . . . . . . . 16 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) → (𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))))
2721, 26ceqsexv 3499 . . . . . . . . . . . . . . 15 (∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
2827rexbii 3110 . . . . . . . . . . . . . 14 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
2920, 28bitr3i 280 . . . . . . . . . . . . 13 (∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
3029rexbii 3110 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
31 r19.41vv 3233 . . . . . . . . . . . . 13 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
3231exbii 1881 . . . . . . . . . . . 12 (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
3319, 30, 323bitr3ri 305 . . . . . . . . . . 11 (∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
3418, 33bitri 278 . . . . . . . . . 10 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
35 eqeq1 2765 . . . . . . . . . . . . 13 (𝑏 = 𝑡 → (𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ↔ 𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))
36352rexbidv 3228 . . . . . . . . . . . 12 (𝑏 = 𝑡 → (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))
3736rexab 3653 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ ∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
38 rexcom4 3290 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
39 rexcom4 3290 . . . . . . . . . . . . . 14 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
40 ovex 7453 . . . . . . . . . . . . . . . 16 (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∈ V
41 oveq2 7428 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) → (𝐴 ·s 𝑡) = (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))
4241oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) → ((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) = ((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
43 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) → (𝑥𝐿 ·s 𝑡) = (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))
4442, 43oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) → (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
4544eqeq2d 2772 . . . . . . . . . . . . . . . 16 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) → (𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))))
4640, 45ceqsexv 3499 . . . . . . . . . . . . . . 15 (∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
4746rexbii 3110 . . . . . . . . . . . . . 14 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
4839, 47bitr3i 280 . . . . . . . . . . . . 13 (∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
4948rexbii 3110 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
50 r19.41vv 3233 . . . . . . . . . . . . 13 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
5150exbii 1881 . . . . . . . . . . . 12 (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
5238, 49, 513bitr3ri 305 . . . . . . . . . . 11 (∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
5337, 52bitri 278 . . . . . . . . . 10 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
5434, 53orbi12i 928 . . . . . . . . 9 ((∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))))
5515, 54bitr2i 279 . . . . . . . 8 ((∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)))
5655rexbii 3110 . . . . . . 7 (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)))
5714, 56bitr3i 280 . . . . . 6 ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)))
5857abbii 2828 . . . . 5 {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))}
5913, 58eqtri 2784 . . . 4 ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))}
60 unab 4254 . . . . 5 ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) = {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))}
61 r19.43 3131 . . . . . . 7 (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))))
62 rexun 4142 . . . . . . . . 9 (∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
63 eqeq1 2765 . . . . . . . . . . . . 13 (𝑏 = 𝑡 → (𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ↔ 𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))
64632rexbidv 3228 . . . . . . . . . . . 12 (𝑏 = 𝑡 → (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))
6564rexab 3653 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ ∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
66 rexcom4 3290 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
67 rexcom4 3290 . . . . . . . . . . . . . 14 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
68 ovex 7453 . . . . . . . . . . . . . . . 16 (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∈ V
69 oveq2 7428 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) → (𝐴 ·s 𝑡) = (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))
7069oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) → ((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) = ((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
71 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) → (𝑥𝑅 ·s 𝑡) = (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))
7270, 71oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) → (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
7372eqeq2d 2772 . . . . . . . . . . . . . . . 16 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) → (𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))))
7468, 73ceqsexv 3499 . . . . . . . . . . . . . . 15 (∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
7574rexbii 3110 . . . . . . . . . . . . . 14 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
7667, 75bitr3i 280 . . . . . . . . . . . . 13 (∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
7776rexbii 3110 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
78 r19.41vv 3233 . . . . . . . . . . . . 13 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
7978exbii 1881 . . . . . . . . . . . 12 (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
8066, 77, 793bitr3ri 305 . . . . . . . . . . 11 (∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
8165, 80bitri 278 . . . . . . . . . 10 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
82 eqeq1 2765 . . . . . . . . . . . . 13 (𝑏 = 𝑡 → (𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ↔ 𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))
83822rexbidv 3228 . . . . . . . . . . . 12 (𝑏 = 𝑡 → (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))
8483rexab 3653 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ ∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
85 rexcom4 3290 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
86 rexcom4 3290 . . . . . . . . . . . . . 14 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
87 ovex 7453 . . . . . . . . . . . . . . . 16 (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∈ V
88 oveq2 7428 . . . . . . . . . . . . . . . . . . 19 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) → (𝐴 ·s 𝑡) = (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))
8988oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) → ((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) = ((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
90 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) → (𝑥𝑅 ·s 𝑡) = (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))
9189, 90oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) → (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
9291eqeq2d 2772 . . . . . . . . . . . . . . . 16 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) → (𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))))
9387, 92ceqsexv 3499 . . . . . . . . . . . . . . 15 (∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
9493rexbii 3110 . . . . . . . . . . . . . 14 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
9586, 94bitr3i 280 . . . . . . . . . . . . 13 (∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
9695rexbii 3110 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
97 r19.41vv 3233 . . . . . . . . . . . . 13 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
9897exbii 1881 . . . . . . . . . . . 12 (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
9985, 96, 983bitr3ri 305 . . . . . . . . . . 11 (∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
10084, 99bitri 278 . . . . . . . . . 10 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
10181, 100orbi12i 928 . . . . . . . . 9 ((∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))))
10262, 101bitr2i 279 . . . . . . . 8 ((∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)))
103102rexbii 3110 . . . . . . 7 (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)))
10461, 103bitr3i 280 . . . . . 6 ((∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)))
105104abbii 2828 . . . . 5 {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))}
10660, 105eqtri 2784 . . . 4 ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))}
10759, 106uneq12i 4113 . . 3 (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))})) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))})
108 unab 4254 . . . . 5 ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))}
109 r19.43 3131 . . . . . . 7 (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))))
110 rexun 4142 . . . . . . . . 9 (∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
11164rexab 3653 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ ∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
112 rexcom4 3290 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
113 rexcom4 3290 . . . . . . . . . . . . . 14 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
11469oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) → ((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) = ((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
115 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) → (𝑥𝐿 ·s 𝑡) = (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))
116114, 115oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) → (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
117116eqeq2d 2772 . . . . . . . . . . . . . . . 16 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) → (𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))))
11868, 117ceqsexv 3499 . . . . . . . . . . . . . . 15 (∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
119118rexbii 3110 . . . . . . . . . . . . . 14 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
120113, 119bitr3i 280 . . . . . . . . . . . . 13 (∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
121120rexbii 3110 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
122 r19.41vv 3233 . . . . . . . . . . . . 13 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
123122exbii 1881 . . . . . . . . . . . 12 (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
124112, 121, 1233bitr3ri 305 . . . . . . . . . . 11 (∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
125111, 124bitri 278 . . . . . . . . . 10 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))))
12683rexab 3653 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ ∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
127 rexcom4 3290 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
128 rexcom4 3290 . . . . . . . . . . . . . 14 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
12988oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) → ((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) = ((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
130 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) → (𝑥𝐿 ·s 𝑡) = (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))
131129, 130oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) → (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
132131eqeq2d 2772 . . . . . . . . . . . . . . . 16 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) → (𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))))
13387, 132ceqsexv 3499 . . . . . . . . . . . . . . 15 (∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
134133rexbii 3110 . . . . . . . . . . . . . 14 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
135128, 134bitr3i 280 . . . . . . . . . . . . 13 (∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
136135rexbii 3110 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
137 r19.41vv 3233 . . . . . . . . . . . . 13 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
138137exbii 1881 . . . . . . . . . . . 12 (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))))
139127, 136, 1383bitr3ri 305 . . . . . . . . . . 11 (∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
140126, 139bitri 278 . . . . . . . . . 10 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))
141125, 140orbi12i 928 . . . . . . . . 9 ((∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))}𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))))
142110, 141bitr2i 279 . . . . . . . 8 ((∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)))
143142rexbii 3110 . . . . . . 7 (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)))
144109, 143bitr3i 280 . . . . . 6 ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡)))
145144abbii 2828 . . . . 5 {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))}
146108, 145eqtri 2784 . . . 4 ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))}
147 unab 4254 . . . . 5 ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) = {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))}
148 r19.43 3131 . . . . . . 7 (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))))
149 rexun 4142 . . . . . . . . 9 (∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
15017rexab 3653 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ ∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
151 rexcom4 3290 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
152 rexcom4 3290 . . . . . . . . . . . . . 14 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
15322oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) → ((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) = ((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
154 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) → (𝑥𝑅 ·s 𝑡) = (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))
155153, 154oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) → (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
156155eqeq2d 2772 . . . . . . . . . . . . . . . 16 (𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) → (𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))))
15721, 156ceqsexv 3499 . . . . . . . . . . . . . . 15 (∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
158157rexbii 3110 . . . . . . . . . . . . . 14 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑡(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
159152, 158bitr3i 280 . . . . . . . . . . . . 13 (∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
160159rexbii 3110 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑡∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
161 r19.41vv 3233 . . . . . . . . . . . . 13 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
162161exbii 1881 . . . . . . . . . . . 12 (∃𝑡∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
163151, 160, 1623bitr3ri 305 . . . . . . . . . . 11 (∃𝑡(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
164150, 163bitri 278 . . . . . . . . . 10 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))))
16536rexab 3653 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ ∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
166 rexcom4 3290 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
167 rexcom4 3290 . . . . . . . . . . . . . 14 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
16841oveq2d 7436 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) → ((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) = ((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
169 oveq2 7428 . . . . . . . . . . . . . . . . . 18 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) → (𝑥𝑅 ·s 𝑡) = (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))
170168, 169oveq12d 7438 . . . . . . . . . . . . . . . . 17 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) → (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
171170eqeq2d 2772 . . . . . . . . . . . . . . . 16 (𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) → (𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))))
17240, 171ceqsexv 3499 . . . . . . . . . . . . . . 15 (∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
173172rexbii 3110 . . . . . . . . . . . . . 14 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑡(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
174167, 173bitr3i 280 . . . . . . . . . . . . 13 (∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
175174rexbii 3110 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑡∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
176 r19.41vv 3233 . . . . . . . . . . . . 13 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
177176exbii 1881 . . . . . . . . . . . 12 (∃𝑡∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))))
178166, 175, 1773bitr3ri 305 . . . . . . . . . . 11 (∃𝑡(∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
179165, 178bitri 278 . . . . . . . . . 10 (∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))
180164, 179orbi12i 928 . . . . . . . . 9 ((∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)) ∨ ∃𝑡 ∈ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))}𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))))
181149, 180bitr2i 279 . . . . . . . 8 ((∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))) ↔ ∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)))
182181rexbii 3110 . . . . . . 7 (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)))
183148, 182bitr3i 280 . . . . . 6 ((∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡)))
184183abbii 2828 . . . . 5 {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))}
185147, 184eqtri 2784 . . . 4 ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))}
186146, 185uneq12i 4113 . . 3 (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))})) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))})
187107, 186oveq12i 7432 . 2 ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}))) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))}) |s ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))})𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝐿 ·s 𝑡))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ ({𝑏 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))} ∪ {𝑏 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))})𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s 𝑡)) -s (𝑥𝑅 ·s 𝑡))}))
18812, 187eqtr4di 2814 1 (𝜑 → (𝐴 ·s (𝐵 ·s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}))))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  ∃wrex 3087   ∪ cun 3897   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420   No csur 27997   <<s cslts 28143   |s ccuts 28145   L cleft 28211   R cright 28212   +s cadds 28345   -s csubs 28406   ·s cmuls 28492
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-muls 28493
This theorem is used by:  mulsass  28552
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