Theorem mulsunif2lem 28612

Description: Lemma for mulsunif2 28613. State the theorem with extra disjoint variable conditions. (Contributed by Scott Fenton, 16-Mar-2025.)

Hypotheses

Ref	Expression
mulsunif2.1	⊢ (𝜑 → 𝐿 <<s 𝑅)
mulsunif2.2	⊢ (𝜑 → 𝑀 <<s 𝑆)
mulsunif2.3	⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
mulsunif2.4	⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))

Assertion

Ref	Expression
mulsunif2lem	⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) −s ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) −s ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)))})))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤   𝐵,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤   𝐿,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤   𝑅,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤   𝑀,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤   𝑆,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤   𝜑,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤

Proof of Theorem mulsunif2lem

Step	Hyp	Ref	Expression
1		mulsunif2.1	. . 3 ⊢ (𝜑 → 𝐿 <<s 𝑅)
2		mulsunif2.2	. . 3 ⊢ (𝜑 → 𝑀 <<s 𝑆)
3		mulsunif2.3	. . 3 ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
4		mulsunif2.4	. . 3 ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))
5	1, 2, 3, 4	mulsunif 28593	. 2 ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))})))
6	1	cutscld 28226	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No)
7	3, 6	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ No)
8	2	cutscld 28226	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 |s 𝑆) ∈ No)
9	4, 8	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ No)
10	7, 9	mulscld 28578	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No)
11	10	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝐴 ·s 𝐵) ∈ No)
12		sltsss1 28208	. . . . . . . . . . . . . . 15 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No)
13	1, 12	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ⊆ No)
14	13	sselda 3913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐿) → 𝑝 ∈ No)
15	14	adantrr 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ∈ No)
16	9	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐵 ∈ No)
17	15, 16	mulscld 28578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝑝 ·s 𝐵) ∈ No)
18	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐴 ∈ No)
19		sltsss1 28208	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆ No)
20	2, 19	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ⊆ No)
21	20	sselda 3913	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑀) → 𝑞 ∈ No)
22	21	adantrl 729	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑞 ∈ No)
23	18, 22	mulscld 28578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝐴 ·s 𝑞) ∈ No)
24	15, 22	mulscld 28578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝑝 ·s 𝑞) ∈ No)
25	23, 24	subscld 28506	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞)) ∈ No)
26	11, 17, 25	subsubs4d 28537	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (((𝐴 ·s 𝐵) −s (𝑝 ·s 𝐵)) −s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞))) = ((𝐴 ·s 𝐵) −s ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞)))))
27	26	oveq2d 7467	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 ·s 𝐵) −s (((𝐴 ·s 𝐵) −s (𝑝 ·s 𝐵)) −s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞)))) = ((𝐴 ·s 𝐵) −s ((𝐴 ·s 𝐵) −s ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞))))))
28	17, 25	addscld 28423	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞))) ∈ No)
29	11, 28	nncansd 28540	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 ·s 𝐵) −s ((𝐴 ·s 𝐵) −s ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞))))) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞))))
30	27, 29	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 ·s 𝐵) −s (((𝐴 ·s 𝐵) −s (𝑝 ·s 𝐵)) −s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞)))) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞))))
31	18, 15	subscld 28506	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝐴 −s 𝑝) ∈ No)
32	31, 16, 22	subsdid 28601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)) = (((𝐴 −s 𝑝) ·s 𝐵) −s ((𝐴 −s 𝑝) ·s 𝑞)))
33	18, 15, 16	subsdird 28602	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 −s 𝑝) ·s 𝐵) = ((𝐴 ·s 𝐵) −s (𝑝 ·s 𝐵)))
34	18, 15, 22	subsdird 28602	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 −s 𝑝) ·s 𝑞) = ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞)))
35	33, 34	oveq12d 7469	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (((𝐴 −s 𝑝) ·s 𝐵) −s ((𝐴 −s 𝑝) ·s 𝑞)) = (((𝐴 ·s 𝐵) −s (𝑝 ·s 𝐵)) −s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞))))
36	32, 35	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)) = (((𝐴 ·s 𝐵) −s (𝑝 ·s 𝐵)) −s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞))))
37	36	oveq2d 7467	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 ·s 𝐵) −s ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞))) = ((𝐴 ·s 𝐵) −s (((𝐴 ·s 𝐵) −s (𝑝 ·s 𝐵)) −s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞)))))
38	17, 23, 24	addsubsassd 28524	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞)) = ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) −s (𝑝 ·s 𝑞))))
39	30, 37, 38	3eqtr4rd 2809	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞)) = ((𝐴 ·s 𝐵) −s ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞))))
40	39	eqeq2d 2774	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞)) ↔ 𝑎 = ((𝐴 ·s 𝐵) −s ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)))))
41	40	2rexbidva 3229	. . . . 5 ⊢ (𝜑 → (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞)) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) −s ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)))))
42	41	abbidv 2829	. . . 4 ⊢ (𝜑 → {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} = {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) −s ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)))})
43	10	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝐴 ·s 𝐵) ∈ No)
44		sltsss2 28209	. . . . . . . . . . . . . . 15 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No)
45	1, 44	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ⊆ No)
46	45	sselda 3913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ No)
47	46	adantrr 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑟 ∈ No)
48		sltsss2 28209	. . . . . . . . . . . . . . 15 ⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆ No)
49	2, 48	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ No)
50	49	sselda 3913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ No)
51	50	adantrl 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑠 ∈ No)
52	47, 51	mulscld 28578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 ·s 𝑠) ∈ No)
53	7	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ∈ No)
54	53, 51	mulscld 28578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝐴 ·s 𝑠) ∈ No)
55	52, 54	subscld 28506	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠)) ∈ No)
56	9	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐵 ∈ No)
57	47, 56	mulscld 28578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 ·s 𝐵) ∈ No)
58	57, 43	subscld 28506	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵)) ∈ No)
59	43, 55, 58	subsubs2d 28538	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝐴 ·s 𝐵) −s (((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠)) −s ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵)))) = ((𝐴 ·s 𝐵) +s (((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵)) −s ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠)))))
60	43, 58, 55	addsubsassd 28524	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (((𝐴 ·s 𝐵) +s ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵))) −s ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠))) = ((𝐴 ·s 𝐵) +s (((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵)) −s ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠)))))
61		pncan3s 28516	. . . . . . . . . . 11 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ (𝑟 ·s 𝐵) ∈ No) → ((𝐴 ·s 𝐵) +s ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵))) = (𝑟 ·s 𝐵))
62	43, 57, 61	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝐴 ·s 𝐵) +s ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵))) = (𝑟 ·s 𝐵))
63	62	oveq1d 7466	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (((𝐴 ·s 𝐵) +s ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵))) −s ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠))) = ((𝑟 ·s 𝐵) −s ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠))))
64	59, 60, 63	3eqtr2d 2804	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝐴 ·s 𝐵) −s (((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠)) −s ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵)))) = ((𝑟 ·s 𝐵) −s ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠))))
65	47, 53	subscld 28506	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 −s 𝐴) ∈ No)
66	65, 51, 56	subsdid 28601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)) = (((𝑟 −s 𝐴) ·s 𝑠) −s ((𝑟 −s 𝐴) ·s 𝐵)))
67	47, 53, 51	subsdird 28602	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 −s 𝐴) ·s 𝑠) = ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠)))
68	47, 53, 56	subsdird 28602	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 −s 𝐴) ·s 𝐵) = ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵)))
69	67, 68	oveq12d 7469	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (((𝑟 −s 𝐴) ·s 𝑠) −s ((𝑟 −s 𝐴) ·s 𝐵)) = (((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠)) −s ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵))))
70	66, 69	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)) = (((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠)) −s ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵))))
71	70	oveq2d 7467	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝐴 ·s 𝐵) −s ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵))) = ((𝐴 ·s 𝐵) −s (((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠)) −s ((𝑟 ·s 𝐵) −s (𝐴 ·s 𝐵)))))
72	57, 54, 52	addsubsassd 28524	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠)) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) −s (𝑟 ·s 𝑠))))
73	57, 52, 54	subsubs2d 28538	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 ·s 𝐵) −s ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠))) = ((𝑟 ·s 𝐵) +s ((𝐴 ·s 𝑠) −s (𝑟 ·s 𝑠))))
74	72, 73	eqtr4d 2801	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠)) = ((𝑟 ·s 𝐵) −s ((𝑟 ·s 𝑠) −s (𝐴 ·s 𝑠))))
75	64, 71, 74	3eqtr4rd 2809	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠)) = ((𝐴 ·s 𝐵) −s ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵))))
76	75	eqeq2d 2774	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠)) ↔ 𝑏 = ((𝐴 ·s 𝐵) −s ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)))))
77	76	2rexbidva 3229	. . . . 5 ⊢ (𝜑 → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠)) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) −s ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)))))
78	77	abbidv 2829	. . . 4 ⊢ (𝜑 → {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))} = {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) −s ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)))})
79	42, 78	uneq12d 4171	. . 3 ⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))}) = ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) −s ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) −s ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)))}))
80	7	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐴 ∈ No)
81	49	sselda 3913	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆) → 𝑢 ∈ No)
82	81	adantrl 729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ No)
83	80, 82	mulscld 28578	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝑢) ∈ No)
84	10	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝐵) ∈ No)
85	83, 84	subscld 28506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) ∈ No)
86	13	sselda 3913	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → 𝑡 ∈ No)
87	86	adantrr 730	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ No)
88	87, 82	mulscld 28578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝑢) ∈ No)
89	9	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ No)
90	87, 89	mulscld 28578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝐵) ∈ No)
91	85, 88, 90	subsubs2d 28538	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) −s ((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵))) = (((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))))
92	90, 88	subscld 28506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢)) ∈ No)
93	83, 92, 84	addsubsd 28525	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))) −s (𝐴 ·s 𝐵)) = (((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))))
94	91, 93	eqtr4d 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) −s ((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵))) = (((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))) −s (𝐴 ·s 𝐵)))
95	94	oveq2d 7467	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 ·s 𝐵) +s (((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) −s ((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵)))) = ((𝐴 ·s 𝐵) +s (((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))) −s (𝐴 ·s 𝐵))))
96	83, 92	addscld 28423	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))) ∈ No)
97		pncan3s 28516	. . . . . . . . . 10 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ ((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))) ∈ No) → ((𝐴 ·s 𝐵) +s (((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))) −s (𝐴 ·s 𝐵))) = ((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))))
98	84, 96, 97	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 ·s 𝐵) +s (((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))) −s (𝐴 ·s 𝐵))) = ((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))))
99	95, 98	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 ·s 𝐵) +s (((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) −s ((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵)))) = ((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))))
100	82, 89	subscld 28506	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑢 −s 𝐵) ∈ No)
101	80, 87, 100	subsdird 28602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)) = ((𝐴 ·s (𝑢 −s 𝐵)) −s (𝑡 ·s (𝑢 −s 𝐵))))
102	80, 82, 89	subsdid 28601	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s (𝑢 −s 𝐵)) = ((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)))
103	87, 82, 89	subsdid 28601	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s (𝑢 −s 𝐵)) = ((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵)))
104	102, 103	oveq12d 7469	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 ·s (𝑢 −s 𝐵)) −s (𝑡 ·s (𝑢 −s 𝐵))) = (((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) −s ((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵))))
105	101, 104	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)) = (((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) −s ((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵))))
106	105	oveq2d 7467	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 ·s 𝐵) +s ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵))) = ((𝐴 ·s 𝐵) +s (((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) −s ((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵)))))
107	90, 83	addscomd 28410	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) = ((𝐴 ·s 𝑢) +s (𝑡 ·s 𝐵)))
108	107	oveq1d 7466	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) = (((𝐴 ·s 𝑢) +s (𝑡 ·s 𝐵)) −s (𝑡 ·s 𝑢)))
109	83, 90, 88	addsubsassd 28524	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝐴 ·s 𝑢) +s (𝑡 ·s 𝐵)) −s (𝑡 ·s 𝑢)) = ((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))))
110	108, 109	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) = ((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))))
111	99, 106, 110	3eqtr4rd 2809	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) = ((𝐴 ·s 𝐵) +s ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵))))
112	111	eqeq2d 2774	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)))))
113	112	2rexbidva 3229	. . . . 5 ⊢ (𝜑 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)))))
114	113	abbidv 2829	. . . 4 ⊢ (𝜑 → {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} = {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)))})
115	45	sselda 3913	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → 𝑣 ∈ No)
116	115	adantrr 730	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ No)
117	9	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ No)
118	116, 117	mulscld 28578	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝐵) ∈ No)
119	20	sselda 3913	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀) → 𝑤 ∈ No)
120	119	adantrl 729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ No)
121	116, 120	mulscld 28578	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝑤) ∈ No)
122	118, 121	subscld 28506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) ∈ No)
123	10	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝐵) ∈ No)
124	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ No)
125	124, 120	mulscld 28578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝑤) ∈ No)
126	122, 123, 125	subsubs2d 28538	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) −s ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤))) = (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s ((𝐴 ·s 𝑤) −s (𝐴 ·s 𝐵))))
127	122, 125, 123	addsubsassd 28524	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)) −s (𝐴 ·s 𝐵)) = (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s ((𝐴 ·s 𝑤) −s (𝐴 ·s 𝐵))))
128	126, 127	eqtr4d 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) −s ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤))) = ((((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)) −s (𝐴 ·s 𝐵)))
129	128	oveq2d 7467	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝐴 ·s 𝐵) +s (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) −s ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤)))) = ((𝐴 ·s 𝐵) +s ((((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)) −s (𝐴 ·s 𝐵))))
130	122, 125	addscld 28423	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)) ∈ No)
131		pncan3s 28516	. . . . . . . . . 10 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)) ∈ No) → ((𝐴 ·s 𝐵) +s ((((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)) −s (𝐴 ·s 𝐵))) = (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)))
132	123, 130, 131	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝐴 ·s 𝐵) +s ((((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)) −s (𝐴 ·s 𝐵))) = (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)))
133	129, 132	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝐴 ·s 𝐵) +s (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) −s ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤)))) = (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)))
134	117, 120	subscld 28506	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐵 −s 𝑤) ∈ No)
135	116, 124, 134	subsdird 28602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)) = ((𝑣 ·s (𝐵 −s 𝑤)) −s (𝐴 ·s (𝐵 −s 𝑤))))
136	116, 117, 120	subsdid 28601	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s (𝐵 −s 𝑤)) = ((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)))
137	124, 117, 120	subsdid 28601	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s (𝐵 −s 𝑤)) = ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤)))
138	136, 137	oveq12d 7469	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ·s (𝐵 −s 𝑤)) −s (𝐴 ·s (𝐵 −s 𝑤))) = (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) −s ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤))))
139	135, 138	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)) = (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) −s ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤))))
140	139	oveq2d 7467	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝐴 ·s 𝐵) +s ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤))) = ((𝐴 ·s 𝐵) +s (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) −s ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤)))))
141	118, 125, 121	addsubsd 28525	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) = (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)))
142	133, 140, 141	3eqtr4rd 2809	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) = ((𝐴 ·s 𝐵) +s ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤))))
143	142	eqeq2d 2774	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)))))
144	143	2rexbidva 3229	. . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)))))
145	144	abbidv 2829	. . . 4 ⊢ (𝜑 → {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))} = {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)))})
146	114, 145	uneq12d 4171	. . 3 ⊢ (𝜑 → ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) = ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)))}))
147	79, 146	oveq12d 7469	. 2 ⊢ (𝜑 → (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))})) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) −s ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) −s ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)))})))
148	5, 147	eqtrd 2798	1 ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) −s ((𝐴 −s 𝑝) ·s (𝐵 −s 𝑞)))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) −s ((𝑟 −s 𝐴) ·s (𝑠 −s 𝐵)))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 −s 𝑡) ·s (𝑢 −s 𝐵)))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 −s 𝐴) ·s (𝐵 −s 𝑤)))})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ⊆ wss 3900   ∪ cun 4152   class class class wbr 5108  (class class class)co 7451  Nocsur 28054   <<s cslts 28200   |s ccuts 28202   +_s cadds 28402   −_s csubs 28463   ·_s cmuls 28549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-muls 28550

This theorem is used by:  mulsunif2  28613