Theorem mulsunif2 28392

Description: Alternate expression for surreal multiplication. Note from [Conway] p. 19. (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
mulsunif2	⊢ (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,𝑝,𝑞   𝐴,𝑏,𝑟,𝑠   𝐴,𝑐,𝑡,𝑢   𝐴,𝑑,𝑣,𝑤   𝐵,𝑎,𝑝,𝑞   𝐵,𝑏,𝑟,𝑠   𝐵,𝑐,𝑡,𝑢   𝐵,𝑑,𝑣,𝑤   𝐿,𝑎,𝑝   𝐿,𝑐,𝑡   𝑀,𝑎,𝑝,𝑞   𝑀,𝑑,𝑣,𝑤   𝑅,𝑏   𝑅,𝑑   𝑅,𝑟   𝑣,𝑅   𝑆,𝑏   𝑆,𝑐   𝑆,𝑟,𝑠   𝑡,𝑆,𝑢

Allowed substitution hints:   𝜑(𝑤, 𝑣, 𝑢, 𝑡, 𝑠, 𝑟, 𝑞, 𝑝, 𝑎, 𝑏, 𝑐, 𝑑)   𝑅(𝑤, 𝑢, 𝑡, 𝑠, 𝑞, 𝑝, 𝑎, 𝑐)   𝑆(𝑤, 𝑣, 𝑞, 𝑝, 𝑎, 𝑑)   𝐿(𝑤, 𝑣, 𝑢, 𝑠, 𝑟, 𝑞, 𝑏, 𝑑)   𝑀(𝑢, 𝑡, 𝑠, 𝑟, 𝑏, 𝑐)

Proof of Theorem mulsunif2

Dummy variables 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 𝑘 𝑙 𝑚 𝑛 𝑜 𝑥 are mutually distinct and distinct from all other variables.

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	mulsunif2lem 28391	. 2 ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} ∪ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))}) |s ({𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} ∪ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))})))
6		eqeq1 2769	. . . . . . 7 ⊢ (𝑒 = 𝑎 → (𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))))
7	6	2rexbidv 3232	. . . . . 6 ⊢ (𝑒 = 𝑎 → (∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))))
8		oveq2 7424	. . . . . . . . . 10 ⊢ (𝑖 = 𝑝 → (𝐴 -s 𝑖) = (𝐴 -s 𝑝))
9	8	oveq1d 7431	. . . . . . . . 9 ⊢ (𝑖 = 𝑝 → ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)) = ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)))
10	9	oveq2d 7432	. . . . . . . 8 ⊢ (𝑖 = 𝑝 → ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))))
11	10	eqeq2d 2776	. . . . . . 7 ⊢ (𝑖 = 𝑝 → (𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)))))
12		oveq2 7424	. . . . . . . . . 10 ⊢ (𝑗 = 𝑞 → (𝐵 -s 𝑗) = (𝐵 -s 𝑞))
13	12	oveq2d 7432	. . . . . . . . 9 ⊢ (𝑗 = 𝑞 → ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗)) = ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))
14	13	oveq2d 7432	. . . . . . . 8 ⊢ (𝑗 = 𝑞 → ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))) = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞))))
15	14	eqeq2d 2776	. . . . . . 7 ⊢ (𝑗 = 𝑞 → (𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑗))) ↔ 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))))
16	11, 15	cbvrex2vw 3250	. . . . . 6 ⊢ (∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞))))
17	7, 16	bitrdi 290	. . . . 5 ⊢ (𝑒 = 𝑎 → (∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗))) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))))
18	17	cbvabv 2835	. . . 4 ⊢ {𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} = {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑝) ·s (𝐵 -s 𝑞)))}
19		eqeq1 2769	. . . . . . 7 ⊢ (𝑓 = 𝑏 → (𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))))
20	19	2rexbidv 3232	. . . . . 6 ⊢ (𝑓 = 𝑏 → (∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))))
21		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑘 = 𝑟 → (𝑘 -s 𝐴) = (𝑟 -s 𝐴))
22	21	oveq1d 7431	. . . . . . . . 9 ⊢ (𝑘 = 𝑟 → ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)) = ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)))
23	22	oveq2d 7432	. . . . . . . 8 ⊢ (𝑘 = 𝑟 → ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))))
24	23	eqeq2d 2776	. . . . . . 7 ⊢ (𝑘 = 𝑟 → (𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)))))
25		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑙 = 𝑠 → (𝑙 -s 𝐵) = (𝑠 -s 𝐵))
26	25	oveq2d 7432	. . . . . . . . 9 ⊢ (𝑙 = 𝑠 → ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵)) = ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))
27	26	oveq2d 7432	. . . . . . . 8 ⊢ (𝑙 = 𝑠 → ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))) = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵))))
28	27	eqeq2d 2776	. . . . . . 7 ⊢ (𝑙 = 𝑠 → (𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))))
29	24, 28	cbvrex2vw 3250	. . . . . 6 ⊢ (∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵))))
30	20, 29	bitrdi 290	. . . . 5 ⊢ (𝑓 = 𝑏 → (∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵))) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))))
31	30	cbvabv 2835	. . . 4 ⊢ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))} = {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = ((𝐴 ·s 𝐵) -s ((𝑟 -s 𝐴) ·s (𝑠 -s 𝐵)))}
32	18, 31	uneq12i 4120	. . 3 ⊢ ({𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} ∪ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -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)))})
33		eqeq1 2769	. . . . . . 7 ⊢ (𝑔 = 𝑐 → (𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))))
34	33	2rexbidv 3232	. . . . . 6 ⊢ (𝑔 = 𝑐 → (∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))))
35		oveq2 7424	. . . . . . . . . 10 ⊢ (𝑚 = 𝑡 → (𝐴 -s 𝑚) = (𝐴 -s 𝑡))
36	35	oveq1d 7431	. . . . . . . . 9 ⊢ (𝑚 = 𝑡 → ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)) = ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)))
37	36	oveq2d 7432	. . . . . . . 8 ⊢ (𝑚 = 𝑡 → ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))))
38	37	eqeq2d 2776	. . . . . . 7 ⊢ (𝑚 = 𝑡 → (𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)))))
39		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑛 = 𝑢 → (𝑛 -s 𝐵) = (𝑢 -s 𝐵))
40	39	oveq2d 7432	. . . . . . . . 9 ⊢ (𝑛 = 𝑢 → ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵)) = ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))
41	40	oveq2d 7432	. . . . . . . 8 ⊢ (𝑛 = 𝑢 → ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))) = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))))
42	41	eqeq2d 2776	. . . . . . 7 ⊢ (𝑛 = 𝑢 → (𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑛 -s 𝐵))) ↔ 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))))
43	38, 42	cbvrex2vw 3250	. . . . . 6 ⊢ (∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵))))
44	34, 43	bitrdi 290	. . . . 5 ⊢ (𝑔 = 𝑐 → (∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))))
45	44	cbvabv 2835	. . . 4 ⊢ {𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} = {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑡) ·s (𝑢 -s 𝐵)))}
46		eqeq1 2769	. . . . . . 7 ⊢ (ℎ = 𝑑 → (ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))))
47	46	2rexbidv 3232	. . . . . 6 ⊢ (ℎ = 𝑑 → (∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))))
48		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑜 = 𝑣 → (𝑜 -s 𝐴) = (𝑣 -s 𝐴))
49	48	oveq1d 7431	. . . . . . . . 9 ⊢ (𝑜 = 𝑣 → ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)) = ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)))
50	49	oveq2d 7432	. . . . . . . 8 ⊢ (𝑜 = 𝑣 → ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))))
51	50	eqeq2d 2776	. . . . . . 7 ⊢ (𝑜 = 𝑣 → (𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)))))
52		oveq2 7424	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝐵 -s 𝑥) = (𝐵 -s 𝑤))
53	52	oveq2d 7432	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥)) = ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))
54	53	oveq2d 7432	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))) = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤))))
55	54	eqeq2d 2776	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))))
56	51, 55	cbvrex2vw 3250	. . . . . 6 ⊢ (∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤))))
57	47, 56	bitrdi 290	. . . . 5 ⊢ (ℎ = 𝑑 → (∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))))
58	57	cbvabv 2835	. . . 4 ⊢ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))} = {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = ((𝐴 ·s 𝐵) +s ((𝑣 -s 𝐴) ·s (𝐵 -s 𝑤)))}
59	45, 58	uneq12i 4120	. . 3 ⊢ ({𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} ∪ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))}) = ({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)))})
60	32, 59	oveq12i 7428	. 2 ⊢ (({𝑒 ∣ ∃𝑖 ∈ 𝐿 ∃𝑗 ∈ 𝑀 𝑒 = ((𝐴 ·s 𝐵) -s ((𝐴 -s 𝑖) ·s (𝐵 -s 𝑗)))} ∪ {𝑓 ∣ ∃𝑘 ∈ 𝑅 ∃𝑙 ∈ 𝑆 𝑓 = ((𝐴 ·s 𝐵) -s ((𝑘 -s 𝐴) ·s (𝑙 -s 𝐵)))}) |s ({𝑔 ∣ ∃𝑚 ∈ 𝐿 ∃𝑛 ∈ 𝑆 𝑔 = ((𝐴 ·s 𝐵) +s ((𝐴 -s 𝑚) ·s (𝑛 -s 𝐵)))} ∪ {ℎ ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 ℎ = ((𝐴 ·s 𝐵) +s ((𝑜 -s 𝐴) ·s (𝐵 -s 𝑥)))})) = (({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)))}))
61	5, 60	eqtrdi 2816	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   = wceq 1570  {cab 2743  ∃wrex 3091   ∪ cun 3904   class class class wbr 5111  (class class class)co 7416   <<s cslts 27979   |s ccuts 27981   +_s cadds 28181   -_s csubs 28242   ·_s cmuls 28328

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-1o 8455  df-2o 8456  df-nadd 8654  df-no 27836  df-lts 27837  df-bday 27838  df-les 27938  df-slts 27980  df-cuts 27982  df-0s 28029  df-made 28049  df-old 28050  df-left 28052  df-right 28053  df-norec 28160  df-norec2 28171  df-adds 28182  df-negs 28243  df-subs 28244  df-muls 28329

This theorem is used by:  remulscl  28724