Theorem mulsasslem2 28559

Description: Lemma for mulsass 28561. 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,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅,𝑧_𝐿,𝑧_𝑅   𝐵,𝑎,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅,𝑧_𝐿,𝑧_𝑅   𝐶,𝑎,𝑥_𝐿,𝑥_𝑅,𝑦_𝐿,𝑦_𝑅,𝑧_𝐿,𝑧_𝑅

Allowed substitution hints:   𝜑(𝑎, 𝑥_𝐿, 𝑥_𝑅, 𝑦_𝐿, 𝑦_𝑅, 𝑧_𝐿, 𝑧_𝑅)

Proof of Theorem mulsasslem2

Dummy variables 𝑏 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lltr 28257	. . . 4 ⊢ (L‘𝐴) <<s (R‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝜑 → (L‘𝐴) <<s (R‘𝐴))
3		mulsasslem.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No)
4		mulsasslem.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ No)
5	3, 4	mulcut2 28528	. . 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 28299	. . . . 5 ⊢ (𝐴 ∈ No → ((L‘𝐴) |s (R‘𝐴)) = 𝐴)
8	6, 7	syl 18	. . . 4 ⊢ (𝜑 → ((L‘𝐴) |s (R‘𝐴)) = 𝐴)
9	8	eqcomd 2766	. . 3 ⊢ (𝜑 → 𝐴 = ((L‘𝐴) |s (R‘𝐴)))
10		mulsval2 28506	. . . 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))})))
11	3, 4, 10	syl2anc 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))})))
12	2, 5, 9, 11	mulsunif 28545	. 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 4252	. . . . 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 3130	. . . . . . 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 4193	. . . . . . . . 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 2764	. . . . . . . . . . . . 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))))
17	16	2rexbidv 3227	. . . . . . . . . . . 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))))
18	17	rexab 3652	. . . . . . . . . . 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 3289	. . . . . . . . . . . 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 3289	. . . . . . . . . . . . . 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 7457	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) −s (𝑦𝐿 ·s 𝑧𝐿)) ∈ V
22		oveq2 7432	. . . . . . . . . . . . . . . . . . 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))))
23	22	oveq2d 7440	. . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . 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))))
25	23, 24	oveq12d 7442	. . . . . . . . . . . . . . . . 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)))))
26	25	eqeq2d 2771	. . . . . . . . . . . . . . . 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))))))
27	21, 26	ceqsexv 3498	. . . . . . . . . . . . . . 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)))))
28	27	rexbii 3109	. . . . . . . . . . . . . 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)))))
29	20, 28	bitr3i 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)))))
30	29	rexbii 3109	. . . . . . . . . . . 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 3232	. . . . . . . . . . . . 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))))
32	31	exbii 1880	. . . . . . . . . . . 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))))
33	19, 30, 32	3bitr3ri 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)))))
34	18, 33	bitri 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 2764	. . . . . . . . . . . . 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))))
36	35	2rexbidv 3227	. . . . . . . . . . . 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))))
37	36	rexab 3652	. . . . . . . . . . 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 3289	. . . . . . . . . . . 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 3289	. . . . . . . . . . . . . 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 7457	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) −s (𝑦𝑅 ·s 𝑧𝑅)) ∈ V
41		oveq2 7432	. . . . . . . . . . . . . . . . . . 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))))
42	41	oveq2d 7440	. . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . 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))))
44	42, 43	oveq12d 7442	. . . . . . . . . . . . . . . . 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)))))
45	44	eqeq2d 2771	. . . . . . . . . . . . . . . 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))))))
46	40, 45	ceqsexv 3498	. . . . . . . . . . . . . . 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)))))
47	46	rexbii 3109	. . . . . . . . . . . . . 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)))))
48	39, 47	bitr3i 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)))))
49	48	rexbii 3109	. . . . . . . . . . . 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 3232	. . . . . . . . . . . . 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))))
51	50	exbii 1880	. . . . . . . . . . . 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))))
52	38, 49, 51	3bitr3ri 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)))))
53	37, 52	bitri 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)))))
54	34, 53	orbi12i 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))))))
55	15, 54	bitr2i 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)))
56	55	rexbii 3109	. . . . . . 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)))
57	14, 56	bitr3i 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)))
58	57	abbii 2827	. . . . 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))}
59	13, 58	eqtri 2783	. . . 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 4252	. . . . 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 3130	. . . . . . 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 4193	. . . . . . . . 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 2764	. . . . . . . . . . . . 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))))
64	63	2rexbidv 3227	. . . . . . . . . . . 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))))
65	64	rexab 3652	. . . . . . . . . . 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 3289	. . . . . . . . . . . 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 3289	. . . . . . . . . . . . . 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 7457	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) −s (𝑦𝐿 ·s 𝑧𝑅)) ∈ V
69		oveq2 7432	. . . . . . . . . . . . . . . . . . 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))))
70	69	oveq2d 7440	. . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . 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))))
72	70, 71	oveq12d 7442	. . . . . . . . . . . . . . . . 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)))))
73	72	eqeq2d 2771	. . . . . . . . . . . . . . . 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))))))
74	68, 73	ceqsexv 3498	. . . . . . . . . . . . . . 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)))))
75	74	rexbii 3109	. . . . . . . . . . . . . 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)))))
76	67, 75	bitr3i 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)))))
77	76	rexbii 3109	. . . . . . . . . . . 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 3232	. . . . . . . . . . . . 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))))
79	78	exbii 1880	. . . . . . . . . . . 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))))
80	66, 77, 79	3bitr3ri 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)))))
81	65, 80	bitri 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 2764	. . . . . . . . . . . . 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))))
83	82	2rexbidv 3227	. . . . . . . . . . . 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))))
84	83	rexab 3652	. . . . . . . . . . 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 3289	. . . . . . . . . . . 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 3289	. . . . . . . . . . . . . 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 7457	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) −s (𝑦𝑅 ·s 𝑧𝐿)) ∈ V
88		oveq2 7432	. . . . . . . . . . . . . . . . . . 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))))
89	88	oveq2d 7440	. . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . 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))))
91	89, 90	oveq12d 7442	. . . . . . . . . . . . . . . . 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)))))
92	91	eqeq2d 2771	. . . . . . . . . . . . . . . 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))))))
93	87, 92	ceqsexv 3498	. . . . . . . . . . . . . . 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)))))
94	93	rexbii 3109	. . . . . . . . . . . . . 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)))))
95	86, 94	bitr3i 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)))))
96	95	rexbii 3109	. . . . . . . . . . . 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 3232	. . . . . . . . . . . . 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))))
98	97	exbii 1880	. . . . . . . . . . . 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))))
99	85, 96, 98	3bitr3ri 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)))))
100	84, 99	bitri 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)))))
101	81, 100	orbi12i 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))))))
102	62, 101	bitr2i 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)))
103	102	rexbii 3109	. . . . . . 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)))
104	61, 103	bitr3i 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)))
105	104	abbii 2827	. . . . 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))}
106	60, 105	eqtri 2783	. . . 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))}
107	59, 106	uneq12i 4164	. . 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 4252	. . . . 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 3130	. . . . . . 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 4193	. . . . . . . . 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))))
111	64	rexab 3652	. . . . . . . . . . 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 3289	. . . . . . . . . . . 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 3289	. . . . . . . . . . . . . 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))))
114	69	oveq2d 7440	. . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . 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))))
116	114, 115	oveq12d 7442	. . . . . . . . . . . . . . . . 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)))))
117	116	eqeq2d 2771	. . . . . . . . . . . . . . . 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))))))
118	68, 117	ceqsexv 3498	. . . . . . . . . . . . . . 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)))))
119	118	rexbii 3109	. . . . . . . . . . . . . 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)))))
120	113, 119	bitr3i 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)))))
121	120	rexbii 3109	. . . . . . . . . . . 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 3232	. . . . . . . . . . . . 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))))
123	122	exbii 1880	. . . . . . . . . . . 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))))
124	112, 121, 123	3bitr3ri 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)))))
125	111, 124	bitri 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)))))
126	83	rexab 3652	. . . . . . . . . . 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 3289	. . . . . . . . . . . 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 3289	. . . . . . . . . . . . . 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))))
129	88	oveq2d 7440	. . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . 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))))
131	129, 130	oveq12d 7442	. . . . . . . . . . . . . . . . 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)))))
132	131	eqeq2d 2771	. . . . . . . . . . . . . . . 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))))))
133	87, 132	ceqsexv 3498	. . . . . . . . . . . . . . 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)))))
134	133	rexbii 3109	. . . . . . . . . . . . . 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)))))
135	128, 134	bitr3i 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)))))
136	135	rexbii 3109	. . . . . . . . . . . 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 3232	. . . . . . . . . . . . 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))))
138	137	exbii 1880	. . . . . . . . . . . 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))))
139	127, 136, 138	3bitr3ri 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)))))
140	126, 139	bitri 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)))))
141	125, 140	orbi12i 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))))))
142	110, 141	bitr2i 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)))
143	142	rexbii 3109	. . . . . . 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)))
144	109, 143	bitr3i 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)))
145	144	abbii 2827	. . . . 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))}
146	108, 145	eqtri 2783	. . . 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 4252	. . . . 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 3130	. . . . . . 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 4193	. . . . . . . . 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))))
150	17	rexab 3652	. . . . . . . . . . 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 3289	. . . . . . . . . . . 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 3289	. . . . . . . . . . . . . 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))))
153	22	oveq2d 7440	. . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . 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))))
155	153, 154	oveq12d 7442	. . . . . . . . . . . . . . . . 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)))))
156	155	eqeq2d 2771	. . . . . . . . . . . . . . . 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))))))
157	21, 156	ceqsexv 3498	. . . . . . . . . . . . . . 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)))))
158	157	rexbii 3109	. . . . . . . . . . . . . 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)))))
159	152, 158	bitr3i 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)))))
160	159	rexbii 3109	. . . . . . . . . . . 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 3232	. . . . . . . . . . . . 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))))
162	161	exbii 1880	. . . . . . . . . . . 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))))
163	151, 160, 162	3bitr3ri 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)))))
164	150, 163	bitri 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)))))
165	36	rexab 3652	. . . . . . . . . . 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 3289	. . . . . . . . . . . 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 3289	. . . . . . . . . . . . . 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))))
168	41	oveq2d 7440	. . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . 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))))
170	168, 169	oveq12d 7442	. . . . . . . . . . . . . . . . 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)))))
171	170	eqeq2d 2771	. . . . . . . . . . . . . . . 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))))))
172	40, 171	ceqsexv 3498	. . . . . . . . . . . . . . 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)))))
173	172	rexbii 3109	. . . . . . . . . . . . . 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)))))
174	167, 173	bitr3i 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)))))
175	174	rexbii 3109	. . . . . . . . . . . 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 3232	. . . . . . . . . . . . 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))))
177	176	exbii 1880	. . . . . . . . . . . 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))))
178	166, 175, 177	3bitr3ri 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)))))
179	165, 178	bitri 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)))))
180	164, 179	orbi12i 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))))))
181	149, 180	bitr2i 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)))
182	181	rexbii 3109	. . . . . . 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)))
183	148, 182	bitr3i 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)))
184	183	abbii 2827	. . . . 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))}
185	147, 184	eqtri 2783	. . . 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))}
186	146, 185	uneq12i 4164	. . 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))})
187	107, 186	oveq12i 7436	. 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))}))
188	12, 187	eqtr4di 2813	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))))}))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ∪ cun 4148   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Nocsur 28006   <<s cslts 28152   |s ccuts 28154  Lcleft 28220  Rcright 28221   +_s cadds 28354   −_s csubs 28415   ·_s cmuls 28501

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-1o 8484  df-2o 8485  df-nadd 8683  df-no 28009  df-lts 28010  df-bday 28011  df-les 28111  df-slts 28153  df-cuts 28155  df-0s 28202  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec 28333  df-norec2 28344  df-adds 28355  df-negs 28416  df-subs 28417  df-muls 28502

This theorem is used by:  mulsass  28561