Theorem addsasslem2 27934

Description: Lemma for addition associativity. Expand the other form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.)

Hypotheses

Ref	Expression
addsasslem.1	⊢ (𝜑 → 𝐴 ∈ No )
addsasslem.2	⊢ (𝜑 → 𝐵 ∈ No )
addsasslem.3	⊢ (𝜑 → 𝐶 ∈ No )

Assertion

Ref	Expression
addsasslem2	⊢ (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧   𝐵,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧   𝐶,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧   𝜑,𝑎,𝑏,𝑐,𝑙,𝑚,𝑛,𝑝,𝑞,𝑟,𝑤,𝑦,𝑧

Proof of Theorem addsasslem2

Dummy variables 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lltropt 27804	. . . 4 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
2	1	a1i 11	. . 3 ⊢ (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
3		addsasslem.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
4		addsasslem.3	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ No )
5	3, 4	addscut 27908	. . . . 5 ⊢ (𝜑 → ((𝐵 +s 𝐶) ∈ No ∧ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s {(𝐵 +s 𝐶)} ∧ {(𝐵 +s 𝐶)} <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})))
6	5	simp2d 1143	. . . 4 ⊢ (𝜑 → ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s {(𝐵 +s 𝐶)})
7	5	simp3d 1144	. . . 4 ⊢ (𝜑 → {(𝐵 +s 𝐶)} <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}))
8		ovex 7386	. . . . . 6 ⊢ (𝐵 +s 𝐶) ∈ V
9	8	snnz 4730	. . . . 5 ⊢ {(𝐵 +s 𝐶)} ≠ ∅
10		sslttr 27736	. . . . 5 ⊢ ((({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s {(𝐵 +s 𝐶)} ∧ {(𝐵 +s 𝐶)} <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}) ∧ {(𝐵 +s 𝐶)} ≠ ∅) → ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}))
11	9, 10	mp3an3 1452	. . . 4 ⊢ ((({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s {(𝐵 +s 𝐶)} ∧ {(𝐵 +s 𝐶)} <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})) → ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}))
12	6, 7, 11	syl2anc 584	. . 3 ⊢ (𝜑 → ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) <<s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}))
13		addsasslem.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ No )
14		lrcut 27836	. . . . 5 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
15	13, 14	syl 17	. . . 4 ⊢ (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
16	15	eqcomd 2735	. . 3 ⊢ (𝜑 → 𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
17		addsval2 27893	. . . 4 ⊢ ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 +s 𝐶) = (({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) |s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})))
18	3, 4, 17	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐵 +s 𝐶) = (({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}) |s ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})))
19	2, 12, 16, 18	addsunif 27932	. 2 ⊢ (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ℎ)}) |s ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)})))
20		rexun 4149	. . . . . . . 8 ⊢ (∃ℎ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ℎ) ↔ (∃ℎ ∈ {𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)}𝑧 = (𝐴 +s ℎ) ∨ ∃ℎ ∈ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}𝑧 = (𝐴 +s ℎ)))
21		eqeq1 2733	. . . . . . . . . . . 12 ⊢ (𝑑 = ℎ → (𝑑 = (𝑚 +s 𝐶) ↔ ℎ = (𝑚 +s 𝐶)))
22	21	rexbidv 3153	. . . . . . . . . . 11 ⊢ (𝑑 = ℎ → (∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶) ↔ ∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝑚 +s 𝐶)))
23	22	rexab 3657	. . . . . . . . . 10 ⊢ (∃ℎ ∈ {𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)}𝑧 = (𝐴 +s ℎ) ↔ ∃ℎ(∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)))
24		rexcom4 3256	. . . . . . . . . . 11 ⊢ (∃𝑚 ∈ ( L ‘𝐵)∃ℎ(ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ ∃ℎ∃𝑚 ∈ ( L ‘𝐵)(ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)))
25		ovex 7386	. . . . . . . . . . . . 13 ⊢ (𝑚 +s 𝐶) ∈ V
26		oveq2 7361	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑚 +s 𝐶) → (𝐴 +s ℎ) = (𝐴 +s (𝑚 +s 𝐶)))
27	26	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑚 +s 𝐶) → (𝑧 = (𝐴 +s ℎ) ↔ 𝑧 = (𝐴 +s (𝑚 +s 𝐶))))
28	25, 27	ceqsexv 3489	. . . . . . . . . . . 12 ⊢ (∃ℎ(ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ 𝑧 = (𝐴 +s (𝑚 +s 𝐶)))
29	28	rexbii 3076	. . . . . . . . . . 11 ⊢ (∃𝑚 ∈ ( L ‘𝐵)∃ℎ(ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)))
30		r19.41v 3159	. . . . . . . . . . . 12 ⊢ (∃𝑚 ∈ ( L ‘𝐵)(ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ (∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)))
31	30	exbii 1848	. . . . . . . . . . 11 ⊢ (∃ℎ∃𝑚 ∈ ( L ‘𝐵)(ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ ∃ℎ(∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)))
32	24, 29, 31	3bitr3ri 302	. . . . . . . . . 10 ⊢ (∃ℎ(∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝑚 +s 𝐶) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)))
33	23, 32	bitri 275	. . . . . . . . 9 ⊢ (∃ℎ ∈ {𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)}𝑧 = (𝐴 +s ℎ) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)))
34		eqeq1 2733	. . . . . . . . . . . 12 ⊢ (𝑒 = ℎ → (𝑒 = (𝐵 +s 𝑛) ↔ ℎ = (𝐵 +s 𝑛)))
35	34	rexbidv 3153	. . . . . . . . . . 11 ⊢ (𝑒 = ℎ → (∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛) ↔ ∃𝑛 ∈ ( L ‘𝐶)ℎ = (𝐵 +s 𝑛)))
36	35	rexab 3657	. . . . . . . . . 10 ⊢ (∃ℎ ∈ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}𝑧 = (𝐴 +s ℎ) ↔ ∃ℎ(∃𝑛 ∈ ( L ‘𝐶)ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)))
37		rexcom4 3256	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ( L ‘𝐶)∃ℎ(ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ ∃ℎ∃𝑛 ∈ ( L ‘𝐶)(ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)))
38		ovex 7386	. . . . . . . . . . . . 13 ⊢ (𝐵 +s 𝑛) ∈ V
39		oveq2 7361	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝐵 +s 𝑛) → (𝐴 +s ℎ) = (𝐴 +s (𝐵 +s 𝑛)))
40	39	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐵 +s 𝑛) → (𝑧 = (𝐴 +s ℎ) ↔ 𝑧 = (𝐴 +s (𝐵 +s 𝑛))))
41	38, 40	ceqsexv 3489	. . . . . . . . . . . 12 ⊢ (∃ℎ(ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ 𝑧 = (𝐴 +s (𝐵 +s 𝑛)))
42	41	rexbii 3076	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ( L ‘𝐶)∃ℎ(ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))
43		r19.41v 3159	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ( L ‘𝐶)(ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ (∃𝑛 ∈ ( L ‘𝐶)ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)))
44	43	exbii 1848	. . . . . . . . . . 11 ⊢ (∃ℎ∃𝑛 ∈ ( L ‘𝐶)(ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ ∃ℎ(∃𝑛 ∈ ( L ‘𝐶)ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)))
45	37, 42, 44	3bitr3ri 302	. . . . . . . . . 10 ⊢ (∃ℎ(∃𝑛 ∈ ( L ‘𝐶)ℎ = (𝐵 +s 𝑛) ∧ 𝑧 = (𝐴 +s ℎ)) ↔ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))
46	36, 45	bitri 275	. . . . . . . . 9 ⊢ (∃ℎ ∈ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}𝑧 = (𝐴 +s ℎ) ↔ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))
47	33, 46	orbi12i 914	. . . . . . . 8 ⊢ ((∃ℎ ∈ {𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)}𝑧 = (𝐴 +s ℎ) ∨ ∃ℎ ∈ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)}𝑧 = (𝐴 +s ℎ)) ↔ (∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)) ∨ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))))
48	20, 47	bitri 275	. . . . . . 7 ⊢ (∃ℎ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ℎ) ↔ (∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)) ∨ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))))
49	48	abbii 2796	. . . . . 6 ⊢ {𝑧 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ℎ)} = {𝑧 ∣ (∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)) ∨ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))}
50		unab 4261	. . . . . 6 ⊢ ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))}) = {𝑧 ∣ (∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶)) ∨ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)))}
51		eqeq1 2733	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝐴 +s (𝐵 +s 𝑛)) ↔ 𝑤 = (𝐴 +s (𝐵 +s 𝑛))))
52	51	rexbidv 3153	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛)) ↔ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))))
53	52	cbvabv 2799	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))} = {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}
54	53	uneq2i 4118	. . . . . 6 ⊢ ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑧 = (𝐴 +s (𝐵 +s 𝑛))}) = ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))})
55	49, 50, 54	3eqtr2i 2758	. . . . 5 ⊢ {𝑧 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ℎ)} = ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))})
56	55	uneq2i 4118	. . . 4 ⊢ ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ℎ)}) = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}))
57		unass 4125	. . . 4 ⊢ (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ ({𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}))
58	56, 57	eqtr4i 2755	. . 3 ⊢ ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ℎ)}) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))})
59		rexun 4149	. . . . . . . 8 ⊢ (∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖) ↔ (∃𝑖 ∈ {𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)}𝑏 = (𝐴 +s 𝑖) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}𝑏 = (𝐴 +s 𝑖)))
60		eqeq1 2733	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑖 → (𝑓 = (𝑞 +s 𝐶) ↔ 𝑖 = (𝑞 +s 𝐶)))
61	60	rexbidv 3153	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑖 → (∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶)))
62	61	rexab 3657	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ {𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)}𝑏 = (𝐴 +s 𝑖) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)))
63		rexcom4 3256	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑖∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)))
64		ovex 7386	. . . . . . . . . . . . 13 ⊢ (𝑞 +s 𝐶) ∈ V
65		oveq2 7361	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑞 +s 𝐶) → (𝐴 +s 𝑖) = (𝐴 +s (𝑞 +s 𝐶)))
66	65	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑞 +s 𝐶) → (𝑏 = (𝐴 +s 𝑖) ↔ 𝑏 = (𝐴 +s (𝑞 +s 𝐶))))
67	64, 66	ceqsexv 3489	. . . . . . . . . . . 12 ⊢ (∃𝑖(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ 𝑏 = (𝐴 +s (𝑞 +s 𝐶)))
68	67	rexbii 3076	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)))
69		r19.41v 3159	. . . . . . . . . . . 12 ⊢ (∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ (∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)))
70	69	exbii 1848	. . . . . . . . . . 11 ⊢ (∃𝑖∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)))
71	63, 68, 70	3bitr3ri 302	. . . . . . . . . 10 ⊢ (∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝑞 +s 𝐶) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)))
72	62, 71	bitri 275	. . . . . . . . 9 ⊢ (∃𝑖 ∈ {𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)}𝑏 = (𝐴 +s 𝑖) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)))
73		eqeq1 2733	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑖 → (𝑔 = (𝐵 +s 𝑟) ↔ 𝑖 = (𝐵 +s 𝑟)))
74	73	rexbidv 3153	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑖 → (∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟)))
75	74	rexab 3657	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}𝑏 = (𝐴 +s 𝑖) ↔ ∃𝑖(∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)))
76		rexcom4 3256	. . . . . . . . . . 11 ⊢ (∃𝑟 ∈ ( R ‘𝐶)∃𝑖(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑖∃𝑟 ∈ ( R ‘𝐶)(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)))
77		ovex 7386	. . . . . . . . . . . . 13 ⊢ (𝐵 +s 𝑟) ∈ V
78		oveq2 7361	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝐵 +s 𝑟) → (𝐴 +s 𝑖) = (𝐴 +s (𝐵 +s 𝑟)))
79	78	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐵 +s 𝑟) → (𝑏 = (𝐴 +s 𝑖) ↔ 𝑏 = (𝐴 +s (𝐵 +s 𝑟))))
80	77, 79	ceqsexv 3489	. . . . . . . . . . . 12 ⊢ (∃𝑖(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ 𝑏 = (𝐴 +s (𝐵 +s 𝑟)))
81	80	rexbii 3076	. . . . . . . . . . 11 ⊢ (∃𝑟 ∈ ( R ‘𝐶)∃𝑖(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))
82		r19.41v 3159	. . . . . . . . . . . 12 ⊢ (∃𝑟 ∈ ( R ‘𝐶)(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ (∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)))
83	82	exbii 1848	. . . . . . . . . . 11 ⊢ (∃𝑖∃𝑟 ∈ ( R ‘𝐶)(𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑖(∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)))
84	76, 81, 83	3bitr3ri 302	. . . . . . . . . 10 ⊢ (∃𝑖(∃𝑟 ∈ ( R ‘𝐶)𝑖 = (𝐵 +s 𝑟) ∧ 𝑏 = (𝐴 +s 𝑖)) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))
85	75, 84	bitri 275	. . . . . . . . 9 ⊢ (∃𝑖 ∈ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}𝑏 = (𝐴 +s 𝑖) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))
86	72, 85	orbi12i 914	. . . . . . . 8 ⊢ ((∃𝑖 ∈ {𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)}𝑏 = (𝐴 +s 𝑖) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)}𝑏 = (𝐴 +s 𝑖)) ↔ (∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)) ∨ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))))
87	59, 86	bitri 275	. . . . . . 7 ⊢ (∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖) ↔ (∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)) ∨ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))))
88	87	abbii 2796	. . . . . 6 ⊢ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)} = {𝑏 ∣ (∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)) ∨ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))}
89		unab 4261	. . . . . 6 ⊢ ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))}) = {𝑏 ∣ (∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶)) ∨ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)))}
90		eqeq1 2733	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (𝑏 = (𝐴 +s (𝐵 +s 𝑟)) ↔ 𝑐 = (𝐴 +s (𝐵 +s 𝑟))))
91	90	rexbidv 3153	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟)) ↔ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))))
92	91	cbvabv 2799	. . . . . . 7 ⊢ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))} = {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}
93	92	uneq2i 4118	. . . . . 6 ⊢ ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑏 = (𝐴 +s (𝐵 +s 𝑟))}) = ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})
94	88, 89, 93	3eqtr2i 2758	. . . . 5 ⊢ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)} = ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})
95	94	uneq2i 4118	. . . 4 ⊢ ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)}) = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}))
96		unass 4125	. . . 4 ⊢ (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}) = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ ({𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}))
97	95, 96	eqtr4i 2755	. . 3 ⊢ ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)}) = (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})
98	58, 97	oveq12i 7365	. 2 ⊢ (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑑 = (𝑚 +s 𝐶)} ∪ {𝑒 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑒 = (𝐵 +s 𝑛)})𝑧 = (𝐴 +s ℎ)}) |s ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑓 = (𝑞 +s 𝐶)} ∪ {𝑔 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑔 = (𝐵 +s 𝑟)})𝑏 = (𝐴 +s 𝑖)})) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))}))
99	19, 98	eqtrdi 2780	1 ⊢ (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s (𝐵 +s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s (𝑚 +s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +s (𝐵 +s 𝑛))}) |s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +s (𝐵 +s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +s (𝑞 +s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +s (𝐵 +s 𝑟))})))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∃wrex 3053   ∪ cun 3903  ∅c0 4286  {csn 4579   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353   No csur 27567   <<s csslt 27709   |s cscut 27711   L cleft 27773   R cright 27774   +_s cadds 27889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-nadd 8591  df-no 27570  df-slt 27571  df-bday 27572  df-sle 27673  df-sslt 27710  df-scut 27712  df-0s 27756  df-made 27775  df-old 27776  df-left 27778  df-right 27779  df-norec2 27879  df-adds 27890

This theorem is referenced by:  addsass  27935