Theorem addsasslem1 27967

Description: Lemma for addition associativity. Expand one 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
addsasslem1	⊢ (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 addsasslem1

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

Step	Hyp	Ref	Expression
1		addsasslem.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
2		addsasslem.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
3	1, 2	addscut 27942	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})))
4	3	simp2d 1143	. . . 4 ⊢ (𝜑 → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)})
5	3	simp3d 1144	. . . 4 ⊢ (𝜑 → {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
6		ovex 7443	. . . . . 6 ⊢ (𝐴 +s 𝐵) ∈ V
7	6	snnz 4757	. . . . 5 ⊢ {(𝐴 +s 𝐵)} ≠ ∅
8		sslttr 27776	. . . . 5 ⊢ ((({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}) ∧ {(𝐴 +s 𝐵)} ≠ ∅) → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
9	7, 8	mp3an3 1452	. . . 4 ⊢ ((({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s {(𝐴 +s 𝐵)} ∧ {(𝐴 +s 𝐵)} <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})) → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
10	4, 5, 9	syl2anc 584	. . 3 ⊢ (𝜑 → ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) <<s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}))
11		lltropt 27841	. . . 4 ⊢ ( L ‘𝐶) <<s ( R ‘𝐶)
12	11	a1i 11	. . 3 ⊢ (𝜑 → ( L ‘𝐶) <<s ( R ‘𝐶))
13		addsval2 27927	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) |s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})))
14	1, 2, 13	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}) |s ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})))
15		addsasslem.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No )
16		lrcut 27872	. . . . 5 ⊢ (𝐶 ∈ No → (( L ‘𝐶) |s ( R ‘𝐶)) = 𝐶)
17	15, 16	syl 17	. . . 4 ⊢ (𝜑 → (( L ‘𝐶) |s ( R ‘𝐶)) = 𝐶)
18	17	eqcomd 2742	. . 3 ⊢ (𝜑 → 𝐶 = (( L ‘𝐶) |s ( R ‘𝐶)))
19	10, 12, 14, 18	addsunif 27966	. 2 ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = (({𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶)} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s ({𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})))
20		unab 4288	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑦 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)}) = {𝑦 ∣ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))}
21		eqeq1 2740	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑧 = ((𝐴 +s 𝑚) +s 𝐶) ↔ 𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
22	21	rexbidv 3165	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
23	22	cbvabv 2806	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)} = {𝑦 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)}
24	23	uneq2i 4145	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)}) = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑦 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)})
25		rexun 4176	. . . . . . 7 ⊢ (∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶) ↔ (∃ℎ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = (ℎ +s 𝐶) ∨ ∃ℎ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = (ℎ +s 𝐶)))
26		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑑 = ℎ → (𝑑 = (𝑙 +s 𝐵) ↔ ℎ = (𝑙 +s 𝐵)))
27	26	rexbidv 3165	. . . . . . . . . 10 ⊢ (𝑑 = ℎ → (∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵) ↔ ∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵)))
28	27	rexab 3683	. . . . . . . . 9 ⊢ (∃ℎ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = (ℎ +s 𝐶) ↔ ∃ℎ(∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)))
29		rexcom4 3273	. . . . . . . . . 10 ⊢ (∃𝑙 ∈ ( L ‘𝐴)∃ℎ(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃ℎ∃𝑙 ∈ ( L ‘𝐴)(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)))
30		ovex 7443	. . . . . . . . . . . 12 ⊢ (𝑙 +s 𝐵) ∈ V
31		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑙 +s 𝐵) → (ℎ +s 𝐶) = ((𝑙 +s 𝐵) +s 𝐶))
32	31	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑙 +s 𝐵) → (𝑦 = (ℎ +s 𝐶) ↔ 𝑦 = ((𝑙 +s 𝐵) +s 𝐶)))
33	30, 32	ceqsexv 3516	. . . . . . . . . . 11 ⊢ (∃ℎ(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ 𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
34	33	rexbii 3084	. . . . . . . . . 10 ⊢ (∃𝑙 ∈ ( L ‘𝐴)∃ℎ(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
35		r19.41v 3175	. . . . . . . . . . 11 ⊢ (∃𝑙 ∈ ( L ‘𝐴)(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ (∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)))
36	35	exbii 1848	. . . . . . . . . 10 ⊢ (∃ℎ∃𝑙 ∈ ( L ‘𝐴)(ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃ℎ(∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)))
37	29, 34, 36	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃ℎ(∃𝑙 ∈ ( L ‘𝐴)ℎ = (𝑙 +s 𝐵) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
38	28, 37	bitri 275	. . . . . . . 8 ⊢ (∃ℎ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = (ℎ +s 𝐶) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶))
39		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑒 = ℎ → (𝑒 = (𝐴 +s 𝑚) ↔ ℎ = (𝐴 +s 𝑚)))
40	39	rexbidv 3165	. . . . . . . . . 10 ⊢ (𝑒 = ℎ → (∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚) ↔ ∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚)))
41	40	rexab 3683	. . . . . . . . 9 ⊢ (∃ℎ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = (ℎ +s 𝐶) ↔ ∃ℎ(∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)))
42		rexcom4 3273	. . . . . . . . . 10 ⊢ (∃𝑚 ∈ ( L ‘𝐵)∃ℎ(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃ℎ∃𝑚 ∈ ( L ‘𝐵)(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)))
43		ovex 7443	. . . . . . . . . . . 12 ⊢ (𝐴 +s 𝑚) ∈ V
44		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐴 +s 𝑚) → (ℎ +s 𝐶) = ((𝐴 +s 𝑚) +s 𝐶))
45	44	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐴 +s 𝑚) → (𝑦 = (ℎ +s 𝐶) ↔ 𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
46	43, 45	ceqsexv 3516	. . . . . . . . . . 11 ⊢ (∃ℎ(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ 𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
47	46	rexbii 3084	. . . . . . . . . 10 ⊢ (∃𝑚 ∈ ( L ‘𝐵)∃ℎ(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
48		r19.41v 3175	. . . . . . . . . . 11 ⊢ (∃𝑚 ∈ ( L ‘𝐵)(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ (∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)))
49	48	exbii 1848	. . . . . . . . . 10 ⊢ (∃ℎ∃𝑚 ∈ ( L ‘𝐵)(ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃ℎ(∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)))
50	42, 47, 49	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃ℎ(∃𝑚 ∈ ( L ‘𝐵)ℎ = (𝐴 +s 𝑚) ∧ 𝑦 = (ℎ +s 𝐶)) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
51	41, 50	bitri 275	. . . . . . . 8 ⊢ (∃ℎ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = (ℎ +s 𝐶) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))
52	38, 51	orbi12i 914	. . . . . . 7 ⊢ ((∃ℎ ∈ {𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)}𝑦 = (ℎ +s 𝐶) ∨ ∃ℎ ∈ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)}𝑦 = (ℎ +s 𝐶)) ↔ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
53	25, 52	bitri 275	. . . . . 6 ⊢ (∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶) ↔ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶)))
54	53	abbii 2803	. . . . 5 ⊢ {𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶)} = {𝑦 ∣ (∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶) ∨ ∃𝑚 ∈ ( L ‘𝐵)𝑦 = ((𝐴 +s 𝑚) +s 𝐶))}
55	20, 24, 54	3eqtr4ri 2770	. . . 4 ⊢ {𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶)} = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +s 𝐵) +s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +s 𝑚) +s 𝐶)})
56	55	uneq1i 4144	. . 3 ⊢ ({𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +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)})
57		unab 4288	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑎 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)}) = {𝑎 ∣ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))}
58		eqeq1 2740	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 = ((𝐴 +s 𝑞) +s 𝐶) ↔ 𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
59	58	rexbidv 3165	. . . . . . 7 ⊢ (𝑏 = 𝑎 → (∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
60	59	cbvabv 2806	. . . . . 6 ⊢ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)} = {𝑎 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)}
61	60	uneq2i 4145	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)}) = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑎 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)})
62		rexun 4176	. . . . . . 7 ⊢ (∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶) ↔ (∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶)))
63		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑖 → (𝑓 = (𝑝 +s 𝐵) ↔ 𝑖 = (𝑝 +s 𝐵)))
64	63	rexbidv 3165	. . . . . . . . . 10 ⊢ (𝑓 = 𝑖 → (∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵)))
65	64	rexab 3683	. . . . . . . . 9 ⊢ (∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑖(∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
66		rexcom4 3273	. . . . . . . . . 10 ⊢ (∃𝑝 ∈ ( R ‘𝐴)∃𝑖(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖∃𝑝 ∈ ( R ‘𝐴)(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
67		ovex 7443	. . . . . . . . . . . 12 ⊢ (𝑝 +s 𝐵) ∈ V
68		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑝 +s 𝐵) → (𝑖 +s 𝐶) = ((𝑝 +s 𝐵) +s 𝐶))
69	68	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑝 +s 𝐵) → (𝑎 = (𝑖 +s 𝐶) ↔ 𝑎 = ((𝑝 +s 𝐵) +s 𝐶)))
70	67, 69	ceqsexv 3516	. . . . . . . . . . 11 ⊢ (∃𝑖(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ 𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
71	70	rexbii 3084	. . . . . . . . . 10 ⊢ (∃𝑝 ∈ ( R ‘𝐴)∃𝑖(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
72		r19.41v 3175	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ ( R ‘𝐴)(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ (∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
73	72	exbii 1848	. . . . . . . . . 10 ⊢ (∃𝑖∃𝑝 ∈ ( R ‘𝐴)(𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖(∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)))
74	66, 71, 73	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑖(∃𝑝 ∈ ( R ‘𝐴)𝑖 = (𝑝 +s 𝐵) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
75	65, 74	bitri 275	. . . . . . . 8 ⊢ (∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶))
76		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑖 → (𝑔 = (𝐴 +s 𝑞) ↔ 𝑖 = (𝐴 +s 𝑞)))
77	76	rexbidv 3165	. . . . . . . . . 10 ⊢ (𝑔 = 𝑖 → (∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞)))
78	77	rexab 3683	. . . . . . . . 9 ⊢ (∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
79		rexcom4 3273	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
80		ovex 7443	. . . . . . . . . . . 12 ⊢ (𝐴 +s 𝑞) ∈ V
81		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝐴 +s 𝑞) → (𝑖 +s 𝐶) = ((𝐴 +s 𝑞) +s 𝐶))
82	81	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝐴 +s 𝑞) → (𝑎 = (𝑖 +s 𝐶) ↔ 𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
83	80, 82	ceqsexv 3516	. . . . . . . . . . 11 ⊢ (∃𝑖(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ 𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
84	83	rexbii 3084	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ ( R ‘𝐵)∃𝑖(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
85		r19.41v 3175	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ (∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
86	85	exbii 1848	. . . . . . . . . 10 ⊢ (∃𝑖∃𝑞 ∈ ( R ‘𝐵)(𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)))
87	79, 84, 86	3bitr3ri 302	. . . . . . . . 9 ⊢ (∃𝑖(∃𝑞 ∈ ( R ‘𝐵)𝑖 = (𝐴 +s 𝑞) ∧ 𝑎 = (𝑖 +s 𝐶)) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
88	78, 87	bitri 275	. . . . . . . 8 ⊢ (∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶) ↔ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))
89	75, 88	orbi12i 914	. . . . . . 7 ⊢ ((∃𝑖 ∈ {𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)}𝑎 = (𝑖 +s 𝐶) ∨ ∃𝑖 ∈ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)}𝑎 = (𝑖 +s 𝐶)) ↔ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
90	62, 89	bitri 275	. . . . . 6 ⊢ (∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶) ↔ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶)))
91	90	abbii 2803	. . . . 5 ⊢ {𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} = {𝑎 ∣ (∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶) ∨ ∃𝑞 ∈ ( R ‘𝐵)𝑎 = ((𝐴 +s 𝑞) +s 𝐶))}
92	57, 61, 91	3eqtr4ri 2770	. . . 4 ⊢ {𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} = ({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +s 𝐵) +s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +s 𝑞) +s 𝐶)})
93	92	uneq1i 4144	. . 3 ⊢ ({𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +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)})
94	56, 93	oveq12i 7422	. 2 ⊢ (({𝑦 ∣ ∃ℎ ∈ ({𝑑 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑑 = (𝑙 +s 𝐵)} ∪ {𝑒 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑒 = (𝐴 +s 𝑚)})𝑦 = (ℎ +s 𝐶)} ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +s 𝐵) +s 𝑛)}) |s ({𝑎 ∣ ∃𝑖 ∈ ({𝑓 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑓 = (𝑝 +s 𝐵)} ∪ {𝑔 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑔 = (𝐴 +s 𝑞)})𝑎 = (𝑖 +s 𝐶)} ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +s 𝐵) +s 𝑟)})) = ((({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)}))
95	19, 94	eqtrdi 2787	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 2714   ≠ wne 2933  ∃wrex 3061   ∪ cun 3929  ∅c0 4313  {csn 4606   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410   No csur 27608   <<s csslt 27749   |s cscut 27751   L cleft 27810   R cright 27811   +_s cadds 27923

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-1o 8485  df-2o 8486  df-nadd 8683  df-no 27611  df-slt 27612  df-bday 27613  df-sle 27714  df-sslt 27750  df-scut 27752  df-0s 27793  df-made 27812  df-old 27813  df-left 27815  df-right 27816  df-norec2 27913  df-adds 27924

This theorem is referenced by:  addsass  27969