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Theorem addsdilem1 27535
Description: Lemma for surreal distribution. Expand the left hand side of the main expression. (Contributed by Scott Fenton, 8-Mar-2025.)
Hypotheses
Ref Expression
addsdilem.1 (𝜑𝐴 No )
addsdilem.2 (𝜑𝐵 No )
addsdilem.3 (𝜑𝐶 No )
Assertion
Ref Expression
addsdilem1 (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}))))
Distinct variable groups:   𝐴,𝑎,𝑥𝐿   𝐴,𝑥𝑅,𝑦𝐿   𝐴,𝑦𝑅   𝐴,𝑧𝐿   𝐴,𝑧𝑅   𝐵,𝑎,𝑥𝐿   𝐵,𝑥𝑅,𝑦𝐿   𝐵,𝑦𝑅   𝐵,𝑧𝐿   𝐵,𝑧𝑅   𝐶,𝑎,𝑥𝐿   𝐶,𝑥𝑅,𝑦𝐿   𝐶,𝑦𝑅   𝐶,𝑧𝐿   𝐶,𝑧𝑅   𝑎,𝑥𝑅,𝑦𝐿   𝑎,𝑦𝑅   𝑎,𝑧𝐿   𝑎,𝑧𝑅   𝑥𝐿,𝑦𝐿   𝑥𝐿,𝑦𝑅   𝑥𝐿,𝑧𝐿   𝑥𝐿,𝑧𝑅   𝑥𝑅,𝑦𝑅   𝑥𝑅,𝑧𝐿   𝑥𝑅,𝑧𝑅
Allowed substitution hints:   𝜑(𝑎,𝑥𝐿,𝑥𝑅,𝑦𝐿,𝑦𝑅,𝑧𝐿,𝑧𝑅)

Proof of Theorem addsdilem1
Dummy variables 𝑏 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lltropt 27296 . . . 4 ( L ‘𝐴) <<s ( R ‘𝐴)
21a1i 11 . . 3 (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
3 addsdilem.2 . . . 4 (𝜑𝐵 No )
4 addsdilem.3 . . . 4 (𝜑𝐶 No )
53, 4addscut2 27392 . . 3 (𝜑 → ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}) <<s ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}))
6 addsdilem.1 . . . . 5 (𝜑𝐴 No )
7 lrcut 27326 . . . . 5 (𝐴 No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
86, 7syl 17 . . . 4 (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
98eqcomd 2738 . . 3 (𝜑𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
10 addsval2 27376 . . . 4 ((𝐵 No 𝐶 No ) → (𝐵 +s 𝐶) = (({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}) |s ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})))
113, 4, 10syl2anc 584 . . 3 (𝜑 → (𝐵 +s 𝐶) = (({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}) |s ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})))
122, 5, 9, 11mulsunif 27534 . 2 (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}) |s ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))})))
13 unab 4295 . . . . 5 ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))}
14 r19.43 3122 . . . . . . 7 (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))))
15 rexun 4187 . . . . . . . . 9 (∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
16 eqeq1 2736 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (𝑡 = (𝑦𝐿 +s 𝐶) ↔ 𝑏 = (𝑦𝐿 +s 𝐶)))
1716rexbidv 3178 . . . . . . . . . . . 12 (𝑡 = 𝑏 → (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶)))
1817rexab 3687 . . . . . . . . . . 11 (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
19 rexcom4 3285 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
20 ovex 7427 . . . . . . . . . . . . . 14 (𝑦𝐿 +s 𝐶) ∈ V
21 oveq2 7402 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑦𝐿 +s 𝐶) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝑦𝐿 +s 𝐶)))
2221oveq2d 7410 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑦𝐿 +s 𝐶) → ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))))
23 oveq2 7402 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))
2422, 23oveq12d 7412 . . . . . . . . . . . . . . 15 (𝑏 = (𝑦𝐿 +s 𝐶) → (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
2524eqeq2d 2743 . . . . . . . . . . . . . 14 (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))))
2620, 25ceqsexv 3523 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
2726rexbii 3094 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
28 r19.41v 3188 . . . . . . . . . . . . 13 (∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
2928exbii 1850 . . . . . . . . . . . 12 (∃𝑏𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
3019, 27, 293bitr3ri 301 . . . . . . . . . . 11 (∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
3118, 30bitri 274 . . . . . . . . . 10 (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))))
32 eqeq1 2736 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (𝑡 = (𝐵 +s 𝑧𝐿) ↔ 𝑏 = (𝐵 +s 𝑧𝐿)))
3332rexbidv 3178 . . . . . . . . . . . 12 (𝑡 = 𝑏 → (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿)))
3433rexab 3687 . . . . . . . . . . 11 (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
35 rexcom4 3285 . . . . . . . . . . . 12 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
36 ovex 7427 . . . . . . . . . . . . . 14 (𝐵 +s 𝑧𝐿) ∈ V
37 oveq2 7402 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝐵 +s 𝑧𝐿) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝐵 +s 𝑧𝐿)))
3837oveq2d 7410 . . . . . . . . . . . . . . . 16 (𝑏 = (𝐵 +s 𝑧𝐿) → ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))))
39 oveq2 7402 . . . . . . . . . . . . . . . 16 (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))
4038, 39oveq12d 7412 . . . . . . . . . . . . . . 15 (𝑏 = (𝐵 +s 𝑧𝐿) → (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
4140eqeq2d 2743 . . . . . . . . . . . . . 14 (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))))
4236, 41ceqsexv 3523 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
4342rexbii 3094 . . . . . . . . . . . 12 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
44 r19.41v 3188 . . . . . . . . . . . . 13 (∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
4544exbii 1850 . . . . . . . . . . . 12 (∃𝑏𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
4635, 43, 453bitr3ri 301 . . . . . . . . . . 11 (∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
4734, 46bitri 274 . . . . . . . . . 10 (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))
4831, 47orbi12i 913 . . . . . . . . 9 ((∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))))
4915, 48bitr2i 275 . . . . . . . 8 ((∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
5049rexbii 3094 . . . . . . 7 (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
5114, 50bitr3i 276 . . . . . 6 ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
5251abbii 2802 . . . . 5 {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿))))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))}
5313, 52eqtri 2760 . . . 4 ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))}
54 unab 4295 . . . . 5 ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))}) = {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))}
55 r19.43 3122 . . . . . . 7 (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))))
56 rexun 4187 . . . . . . . . 9 (∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
57 eqeq1 2736 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (𝑡 = (𝑦𝑅 +s 𝐶) ↔ 𝑏 = (𝑦𝑅 +s 𝐶)))
5857rexbidv 3178 . . . . . . . . . . . 12 (𝑡 = 𝑏 → (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶)))
5958rexab 3687 . . . . . . . . . . 11 (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
60 rexcom4 3285 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
61 ovex 7427 . . . . . . . . . . . . . 14 (𝑦𝑅 +s 𝐶) ∈ V
62 oveq2 7402 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑦𝑅 +s 𝐶) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝑦𝑅 +s 𝐶)))
6362oveq2d 7410 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑦𝑅 +s 𝐶) → ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))))
64 oveq2 7402 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))
6563, 64oveq12d 7412 . . . . . . . . . . . . . . 15 (𝑏 = (𝑦𝑅 +s 𝐶) → (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
6665eqeq2d 2743 . . . . . . . . . . . . . 14 (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))))
6761, 66ceqsexv 3523 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
6867rexbii 3094 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
69 r19.41v 3188 . . . . . . . . . . . . 13 (∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
7069exbii 1850 . . . . . . . . . . . 12 (∃𝑏𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
7160, 68, 703bitr3ri 301 . . . . . . . . . . 11 (∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
7259, 71bitri 274 . . . . . . . . . 10 (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))))
73 eqeq1 2736 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (𝑡 = (𝐵 +s 𝑧𝑅) ↔ 𝑏 = (𝐵 +s 𝑧𝑅)))
7473rexbidv 3178 . . . . . . . . . . . 12 (𝑡 = 𝑏 → (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅)))
7574rexab 3687 . . . . . . . . . . 11 (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
76 rexcom4 3285 . . . . . . . . . . . 12 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
77 ovex 7427 . . . . . . . . . . . . . 14 (𝐵 +s 𝑧𝑅) ∈ V
78 oveq2 7402 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝐵 +s 𝑧𝑅) → (𝐴 ·s 𝑏) = (𝐴 ·s (𝐵 +s 𝑧𝑅)))
7978oveq2d 7410 . . . . . . . . . . . . . . . 16 (𝑏 = (𝐵 +s 𝑧𝑅) → ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))))
80 oveq2 7402 . . . . . . . . . . . . . . . 16 (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))
8179, 80oveq12d 7412 . . . . . . . . . . . . . . 15 (𝑏 = (𝐵 +s 𝑧𝑅) → (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
8281eqeq2d 2743 . . . . . . . . . . . . . 14 (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))))
8377, 82ceqsexv 3523 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
8483rexbii 3094 . . . . . . . . . . . 12 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
85 r19.41v 3188 . . . . . . . . . . . . 13 (∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
8685exbii 1850 . . . . . . . . . . . 12 (∃𝑏𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
8776, 84, 863bitr3ri 301 . . . . . . . . . . 11 (∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
8875, 87bitri 274 . . . . . . . . . 10 (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))
8972, 88orbi12i 913 . . . . . . . . 9 ((∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))))
9056, 89bitr2i 275 . . . . . . . 8 ((∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
9190rexbii 3094 . . . . . . 7 (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
9255, 91bitr3i 276 . . . . . 6 ((∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
9392abbii 2802 . . . . 5 {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅))))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}
9454, 93eqtri 2760 . . . 4 ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))}) = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}
9553, 94uneq12i 4158 . . 3 (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))})
96 unab 4295 . . . . 5 ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) = {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))}
97 r19.43 3122 . . . . . . 7 (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))) ↔ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))))
98 rexun 4187 . . . . . . . . 9 (∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
9958rexab 3687 . . . . . . . . . . 11 (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
100 rexcom4 3285 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
10162oveq2d 7410 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑦𝑅 +s 𝐶) → ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))))
102 oveq2 7402 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))
103101, 102oveq12d 7412 . . . . . . . . . . . . . . 15 (𝑏 = (𝑦𝑅 +s 𝐶) → (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
104103eqeq2d 2743 . . . . . . . . . . . . . 14 (𝑏 = (𝑦𝑅 +s 𝐶) → (𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))))
10561, 104ceqsexv 3523 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
106105rexbii 3094 . . . . . . . . . . . 12 (∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑏(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
107 r19.41v 3188 . . . . . . . . . . . . 13 (∃𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
108107exbii 1850 . . . . . . . . . . . 12 (∃𝑏𝑦𝑅 ∈ ( R ‘𝐵)(𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
109100, 106, 1083bitr3ri 301 . . . . . . . . . . 11 (∃𝑏(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑏 = (𝑦𝑅 +s 𝐶) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
11099, 109bitri 274 . . . . . . . . . 10 (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))))
11174rexab 3687 . . . . . . . . . . 11 (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
112 rexcom4 3285 . . . . . . . . . . . 12 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
11378oveq2d 7410 . . . . . . . . . . . . . . . 16 (𝑏 = (𝐵 +s 𝑧𝑅) → ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))))
114 oveq2 7402 . . . . . . . . . . . . . . . 16 (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑥𝐿 ·s 𝑏) = (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))
115113, 114oveq12d 7412 . . . . . . . . . . . . . . 15 (𝑏 = (𝐵 +s 𝑧𝑅) → (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
116115eqeq2d 2743 . . . . . . . . . . . . . 14 (𝑏 = (𝐵 +s 𝑧𝑅) → (𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))))
11777, 116ceqsexv 3523 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
118117rexbii 3094 . . . . . . . . . . . 12 (∃𝑧𝑅 ∈ ( R ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
119 r19.41v 3188 . . . . . . . . . . . . 13 (∃𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
120119exbii 1850 . . . . . . . . . . . 12 (∃𝑏𝑧𝑅 ∈ ( R ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))))
121112, 118, 1203bitr3ri 301 . . . . . . . . . . 11 (∃𝑏(∃𝑧𝑅 ∈ ( R ‘𝐶)𝑏 = (𝐵 +s 𝑧𝑅) ∧ 𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
122111, 121bitri 274 . . . . . . . . . 10 (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))
123110, 122orbi12i 913 . . . . . . . . 9 ((∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)}𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))) ↔ (∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))))
12498, 123bitr2i 275 . . . . . . . 8 ((∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
125124rexbii 3094 . . . . . . 7 (∃𝑥𝐿 ∈ ( L ‘𝐴)(∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
12697, 125bitr3i 276 . . . . . 6 ((∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏)))
127126abbii 2802 . . . . 5 {𝑎 ∣ (∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶))) ∨ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅))))} = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))}
12896, 127eqtri 2760 . . . 4 ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) = {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))}
129 unab 4295 . . . . 5 ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}) = {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))}
130 r19.43 3122 . . . . . . 7 (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))))
131 rexun 4187 . . . . . . . . 9 (∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
13217rexab 3687 . . . . . . . . . . 11 (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
133 rexcom4 3285 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
13421oveq2d 7410 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑦𝐿 +s 𝐶) → ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))))
135 oveq2 7402 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))
136134, 135oveq12d 7412 . . . . . . . . . . . . . . 15 (𝑏 = (𝑦𝐿 +s 𝐶) → (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
137136eqeq2d 2743 . . . . . . . . . . . . . 14 (𝑏 = (𝑦𝐿 +s 𝐶) → (𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))))
13820, 137ceqsexv 3523 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
139138rexbii 3094 . . . . . . . . . . . 12 (∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑏(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
140 r19.41v 3188 . . . . . . . . . . . . 13 (∃𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
141140exbii 1850 . . . . . . . . . . . 12 (∃𝑏𝑦𝐿 ∈ ( L ‘𝐵)(𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
142133, 139, 1413bitr3ri 301 . . . . . . . . . . 11 (∃𝑏(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑏 = (𝑦𝐿 +s 𝐶) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
143132, 142bitri 274 . . . . . . . . . 10 (∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))))
14433rexab 3687 . . . . . . . . . . 11 (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
145 rexcom4 3285 . . . . . . . . . . . 12 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
14637oveq2d 7410 . . . . . . . . . . . . . . . 16 (𝑏 = (𝐵 +s 𝑧𝐿) → ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) = ((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))))
147 oveq2 7402 . . . . . . . . . . . . . . . 16 (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑥𝑅 ·s 𝑏) = (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))
148146, 147oveq12d 7412 . . . . . . . . . . . . . . 15 (𝑏 = (𝐵 +s 𝑧𝐿) → (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
149148eqeq2d 2743 . . . . . . . . . . . . . 14 (𝑏 = (𝐵 +s 𝑧𝐿) → (𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))))
15036, 149ceqsexv 3523 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
151150rexbii 3094 . . . . . . . . . . . 12 (∃𝑧𝐿 ∈ ( L ‘𝐶)∃𝑏(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
152 r19.41v 3188 . . . . . . . . . . . . 13 (∃𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
153152exbii 1850 . . . . . . . . . . . 12 (∃𝑏𝑧𝐿 ∈ ( L ‘𝐶)(𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))))
154145, 151, 1533bitr3ri 301 . . . . . . . . . . 11 (∃𝑏(∃𝑧𝐿 ∈ ( L ‘𝐶)𝑏 = (𝐵 +s 𝑧𝐿) ∧ 𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
155144, 154bitri 274 . . . . . . . . . 10 (∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ↔ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))
156143, 155orbi12i 913 . . . . . . . . 9 ((∃𝑏 ∈ {𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)) ∨ ∃𝑏 ∈ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)}𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))) ↔ (∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))))
157131, 156bitr2i 275 . . . . . . . 8 ((∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
158157rexbii 3094 . . . . . . 7 (∃𝑥𝑅 ∈ ( R ‘𝐴)(∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
159130, 158bitr3i 276 . . . . . 6 ((∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏)))
160159abbii 2802 . . . . 5 {𝑎 ∣ (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶))) ∨ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿))))} = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}
161129, 160eqtri 2760 . . . 4 ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}) = {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}
162128, 161uneq12i 4158 . . 3 (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))})) = ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))})
16395, 162oveq12i 7406 . 2 ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}))) = (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}) |s ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝑅 ∈ ( R ‘𝐵)𝑡 = (𝑦𝑅 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐶)𝑡 = (𝐵 +s 𝑧𝑅)})𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝐿 ·s 𝑏))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑏 ∈ ({𝑡 ∣ ∃𝑦𝐿 ∈ ( L ‘𝐵)𝑡 = (𝑦𝐿 +s 𝐶)} ∪ {𝑡 ∣ ∃𝑧𝐿 ∈ ( L ‘𝐶)𝑡 = (𝐵 +s 𝑧𝐿)})𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s 𝑏)) -s (𝑥𝑅 ·s 𝑏))}))
16412, 163eqtr4di 2790 1 (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wrex 3070  cun 3943   class class class wbr 5142  cfv 6533  (class class class)co 7394   No csur 27072   <<s csslt 27211   |s cscut 27213   L cleft 27269   R cright 27270   +s cadds 27372   -s csubs 27424   ·s cmuls 27491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-1o 8450  df-2o 8451  df-nadd 8650  df-no 27075  df-slt 27076  df-bday 27077  df-sle 27177  df-sslt 27212  df-scut 27214  df-0s 27254  df-made 27271  df-old 27272  df-left 27274  df-right 27275  df-norec 27351  df-norec2 27362  df-adds 27373  df-negs 27425  df-subs 27426  df-muls 27492
This theorem is referenced by:  addsdi  27539
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