Theorem mulsunif2lem 28210

Description: Lemma for mulsunif2 28211. 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 28191	. 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	scutcld 27863	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No )
7	3, 6	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ No )
8	2	scutcld 27863	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 |s 𝑆) ∈ No )
9	4, 8	eqeltrd 2839	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ No )
10	7, 9	mulscld 28176	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
11	10	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝐴 ·s 𝐵) ∈ No )
12		ssltss1 27848	. . . . . . . . . . . . . . 15 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No )
13	1, 12	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ⊆ No )
14	13	sselda 3995	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐿) → 𝑝 ∈ No )
15	14	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ∈ No )
16	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐵 ∈ No )
17	15, 16	mulscld 28176	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝑝 ·s 𝐵) ∈ No )
18	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐴 ∈ No )
19		ssltss1 27848	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆ No )
20	2, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ⊆ No )
21	20	sselda 3995	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝑀) → 𝑞 ∈ No )
22	21	adantrl 716	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑞 ∈ No )
23	18, 22	mulscld 28176	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝐴 ·s 𝑞) ∈ No )
24	15, 22	mulscld 28176	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝑝 ·s 𝑞) ∈ No )
25	23, 24	subscld 28108	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞)) ∈ No )
26	11, 17, 25	subsubs4d 28139	. . . . . . . . . 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 7447	. . . . . . . . 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 28028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝑝 ·s 𝐵) +s ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞))) ∈ No )
29	11, 28	nncansd 28141	. . . . . . . . 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 2775	. . . . . . . 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 28108	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → (𝐴 -s 𝑝) ∈ No )
32	31, 16, 22	subsdid 28199	. . . . . . . . . 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 28200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 -s 𝑝) ·s 𝐵) = ((𝐴 ·s 𝐵) -s (𝑝 ·s 𝐵)))
34	18, 15, 22	subsdird 28200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝐴 -s 𝑝) ·s 𝑞) = ((𝐴 ·s 𝑞) -s (𝑝 ·s 𝑞)))
35	33, 34	oveq12d 7449	. . . . . . . . . 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 2775	. . . . . . . . 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 7447	. . . . . . . 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 28126	. . . . . . . 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 2786	. . . . . . 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 2746	. . . . . 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 3218	. . . . 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 2806	. . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝐴 ·s 𝐵) ∈ No )
44		ssltss2 27849	. . . . . . . . . . . . . . 15 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No )
45	1, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ⊆ No )
46	45	sselda 3995	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ No )
47	46	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑟 ∈ No )
48		ssltss2 27849	. . . . . . . . . . . . . . 15 ⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆ No )
49	2, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ No )
50	49	sselda 3995	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ No )
51	50	adantrl 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑠 ∈ No )
52	47, 51	mulscld 28176	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 ·s 𝑠) ∈ No )
53	7	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ∈ No )
54	53, 51	mulscld 28176	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝐴 ·s 𝑠) ∈ No )
55	52, 54	subscld 28108	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 ·s 𝑠) -s (𝐴 ·s 𝑠)) ∈ No )
56	9	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐵 ∈ No )
57	47, 56	mulscld 28176	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 ·s 𝐵) ∈ No )
58	57, 43	subscld 28108	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 ·s 𝐵) -s (𝐴 ·s 𝐵)) ∈ No )
59	43, 55, 58	subsubs2d 28140	. . . . . . . . 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 28126	. . . . . . . . 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 28118	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝐴 ·s 𝐵) +s ((𝑟 ·s 𝐵) -s (𝐴 ·s 𝐵))) = (𝑟 ·s 𝐵))
63	62	oveq1d 7446	. . . . . . . . 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 2781	. . . . . . . 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 28108	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 -s 𝐴) ∈ No )
66	65, 51, 56	subsdid 28199	. . . . . . . . . 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 28200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 -s 𝐴) ·s 𝑠) = ((𝑟 ·s 𝑠) -s (𝐴 ·s 𝑠)))
68	47, 53, 56	subsdird 28200	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 -s 𝐴) ·s 𝐵) = ((𝑟 ·s 𝐵) -s (𝐴 ·s 𝐵)))
69	67, 68	oveq12d 7449	. . . . . . . . . 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 2775	. . . . . . . . 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 7447	. . . . . . . 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 28126	. . . . . . . . 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 28140	. . . . . . . . 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 2778	. . . . . . . 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 2786	. . . . . . 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 2746	. . . . . 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 3218	. . . . 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 2806	. . . 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 4179	. . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐴 ∈ No )
81	49	sselda 3995	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆) → 𝑢 ∈ No )
82	81	adantrl 716	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ No )
83	80, 82	mulscld 28176	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝑢) ∈ No )
84	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝐵) ∈ No )
85	83, 84	subscld 28108	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 ·s 𝑢) -s (𝐴 ·s 𝐵)) ∈ No )
86	13	sselda 3995	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → 𝑡 ∈ No )
87	86	adantrr 717	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ No )
88	87, 82	mulscld 28176	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝑢) ∈ No )
89	9	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ No )
90	87, 89	mulscld 28176	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝐵) ∈ No )
91	85, 88, 90	subsubs2d 28140	. . . . . . . . . . 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 28108	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) ∈ No )
93	83, 92, 84	addsubsd 28127	. . . . . . . . . . 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 2778	. . . . . . . . . 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 7447	. . . . . . . . 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 28028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝐴 ·s 𝑢) +s ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢))) ∈ No )
97		pncan3s 28118	. . . . . . . . . 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 584	. . . . . . . . 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 2775	. . . . . . . 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 28108	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑢 -s 𝐵) ∈ No )
101	80, 87, 100	subsdird 28200	. . . . . . . . . 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 28199	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s (𝑢 -s 𝐵)) = ((𝐴 ·s 𝑢) -s (𝐴 ·s 𝐵)))
103	87, 82, 89	subsdid 28199	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s (𝑢 -s 𝐵)) = ((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)))
104	102, 103	oveq12d 7449	. . . . . . . . . 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 2775	. . . . . . . . 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 7447	. . . . . . . 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 28015	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) = ((𝐴 ·s 𝑢) +s (𝑡 ·s 𝐵)))
108	107	oveq1d 7446	. . . . . . . . 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 28126	. . . . . . . . 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 2775	. . . . . . . 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 2786	. . . . . . 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 2746	. . . . . 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 3218	. . . . 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 2806	. . . 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 3995	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → 𝑣 ∈ No )
116	115	adantrr 717	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ No )
117	9	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ No )
118	116, 117	mulscld 28176	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝐵) ∈ No )
119	20	sselda 3995	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑀) → 𝑤 ∈ No )
120	119	adantrl 716	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ No )
121	116, 120	mulscld 28176	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝑤) ∈ No )
122	118, 121	subscld 28108	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)) ∈ No )
123	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝐵) ∈ No )
124	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ No )
125	124, 120	mulscld 28176	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝑤) ∈ No )
126	122, 123, 125	subsubs2d 28140	. . . . . . . . . . 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 28126	. . . . . . . . . . 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 2778	. . . . . . . . . 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 7447	. . . . . . . . 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 28028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)) ∈ No )
131		pncan3s 28118	. . . . . . . . . 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 584	. . . . . . . . 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 2775	. . . . . . . 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 28108	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐵 -s 𝑤) ∈ No )
135	116, 124, 134	subsdird 28200	. . . . . . . . . 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 28199	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s (𝐵 -s 𝑤)) = ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)))
137	124, 117, 120	subsdid 28199	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s (𝐵 -s 𝑤)) = ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑤)))
138	136, 137	oveq12d 7449	. . . . . . . . . 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 2775	. . . . . . . . 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 7447	. . . . . . . 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 28127	. . . . . . . 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 2786	. . . . . . 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 2746	. . . . . 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 3218	. . . . 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 2806	. . . 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 4179	. . 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 7449	. 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 2775	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)))})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  ∃wrex 3068   ∪ cun 3961   ⊆ wss 3963   class class class wbr 5148  (class class class)co 7431   No csur 27699   <<s csslt 27840   |s cscut 27842   +_s cadds 28007   -_s csubs 28067   ·_s cmuls 28147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148

This theorem is referenced by:  mulsunif2  28211