Theorem addcmpblnq0 7663

Description: Lemma showing compatibility of addition on nonnegative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)

Assertion

Ref	Expression
addcmpblnq0	⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅)) → ⟨((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)⟩ ~Q0 ⟨((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)), (𝐷 ·o 𝑆)⟩))

Proof of Theorem addcmpblnq0

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nndi 6654	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑥 ·o (𝑦 +o 𝑧)) = ((𝑥 ·o 𝑦) +o (𝑥 ·o 𝑧)))
2	1	adantl 277	. . . . . . 7 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → (𝑥 ·o (𝑦 +o 𝑧)) = ((𝑥 ·o 𝑦) +o (𝑥 ·o 𝑧)))
3		simplll 535	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝐴 ∈ ω)
4		simprlr 540	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝐺 ∈ N)
5		pinn 7529	. . . . . . . . 9 ⊢ (𝐺 ∈ N → 𝐺 ∈ ω)
6	4, 5	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝐺 ∈ ω)
7		nnmcl 6649	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐺 ∈ ω) → (𝐴 ·o 𝐺) ∈ ω)
8	3, 6, 7	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (𝐴 ·o 𝐺) ∈ ω)
9		simpllr 536	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝐵 ∈ N)
10		pinn 7529	. . . . . . . . 9 ⊢ (𝐵 ∈ N → 𝐵 ∈ ω)
11	9, 10	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝐵 ∈ ω)
12		simprll 539	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝐹 ∈ ω)
13		nnmcl 6649	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐹 ∈ ω) → (𝐵 ·o 𝐹) ∈ ω)
14	11, 12, 13	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (𝐵 ·o 𝐹) ∈ ω)
15		simplrr 538	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝐷 ∈ N)
16		pinn 7529	. . . . . . . . 9 ⊢ (𝐷 ∈ N → 𝐷 ∈ ω)
17	15, 16	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝐷 ∈ ω)
18		simprrr 542	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝑆 ∈ N)
19		pinn 7529	. . . . . . . . 9 ⊢ (𝑆 ∈ N → 𝑆 ∈ ω)
20	18, 19	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝑆 ∈ ω)
21		nnmcl 6649	. . . . . . . 8 ⊢ ((𝐷 ∈ ω ∧ 𝑆 ∈ ω) → (𝐷 ·o 𝑆) ∈ ω)
22	17, 20, 21	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (𝐷 ·o 𝑆) ∈ ω)
23		nnacl 6648	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +o 𝑦) ∈ ω)
24	23	adantl 277	. . . . . . 7 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 +o 𝑦) ∈ ω)
25		nnmcom 6657	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·o 𝑦) = (𝑦 ·o 𝑥))
26	25	adantl 277	. . . . . . 7 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ·o 𝑦) = (𝑦 ·o 𝑥))
27	2, 8, 14, 22, 24, 26	caovdir2d 6199	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = (((𝐴 ·o 𝐺) ·o (𝐷 ·o 𝑆)) +o ((𝐵 ·o 𝐹) ·o (𝐷 ·o 𝑆))))
28		nnmass 6655	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑥 ·o 𝑦) ·o 𝑧) = (𝑥 ·o (𝑦 ·o 𝑧)))
29	28	adantl 277	. . . . . . . 8 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑥 ·o 𝑦) ·o 𝑧) = (𝑥 ·o (𝑦 ·o 𝑧)))
30		nnmcl 6649	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·o 𝑦) ∈ ω)
31	30	adantl 277	. . . . . . . 8 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ·o 𝑦) ∈ ω)
32	3, 6, 17, 26, 29, 20, 31	caov4d 6207	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → ((𝐴 ·o 𝐺) ·o (𝐷 ·o 𝑆)) = ((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆)))
33	11, 12, 17, 26, 29, 20, 31	caov4d 6207	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → ((𝐵 ·o 𝐹) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆)))
34	32, 33	oveq12d 6036	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·o 𝐺) ·o (𝐷 ·o 𝑆)) +o ((𝐵 ·o 𝐹) ·o (𝐷 ·o 𝑆))) = (((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆))))
35	27, 34	eqtrd 2264	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = (((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆))))
36		oveq1 6025	. . . . . 6 ⊢ ((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) → ((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆)) = ((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)))
37		oveq2 6026	. . . . . 6 ⊢ ((𝐹 ·o 𝑆) = (𝐺 ·o 𝑅) → ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆)) = ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅)))
38	36, 37	oveqan12d 6037	. . . . 5 ⊢ (((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅)) → (((𝐴 ·o 𝐷) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐹 ·o 𝑆))) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅))))
39	35, 38	sylan9eq 2284	. . . 4 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ ((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅))))
40		nnmcl 6649	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐺 ∈ ω) → (𝐵 ·o 𝐺) ∈ ω)
41	11, 6, 40	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (𝐵 ·o 𝐺) ∈ ω)
42		simplrl 537	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝐶 ∈ ω)
43		nnmcl 6649	. . . . . . . 8 ⊢ ((𝐶 ∈ ω ∧ 𝑆 ∈ ω) → (𝐶 ·o 𝑆) ∈ ω)
44	42, 20, 43	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (𝐶 ·o 𝑆) ∈ ω)
45		simprrl 541	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → 𝑅 ∈ ω)
46		nnmcl 6649	. . . . . . . 8 ⊢ ((𝐷 ∈ ω ∧ 𝑅 ∈ ω) → (𝐷 ·o 𝑅) ∈ ω)
47	17, 45, 46	syl2anc 411	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (𝐷 ·o 𝑅) ∈ ω)
48		nndi 6654	. . . . . . 7 ⊢ (((𝐵 ·o 𝐺) ∈ ω ∧ (𝐶 ·o 𝑆) ∈ ω ∧ (𝐷 ·o 𝑅) ∈ ω) → ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐺) ·o (𝐶 ·o 𝑆)) +o ((𝐵 ·o 𝐺) ·o (𝐷 ·o 𝑅))))
49	41, 44, 47, 48	syl3anc 1273	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐺) ·o (𝐶 ·o 𝑆)) +o ((𝐵 ·o 𝐺) ·o (𝐷 ·o 𝑅))))
50	11, 6, 42, 26, 29, 20, 31	caov4d 6207	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → ((𝐵 ·o 𝐺) ·o (𝐶 ·o 𝑆)) = ((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)))
51	11, 6, 17, 26, 29, 45, 31	caov4d 6207	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → ((𝐵 ·o 𝐺) ·o (𝐷 ·o 𝑅)) = ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅)))
52	50, 51	oveq12d 6036	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐵 ·o 𝐺) ·o (𝐶 ·o 𝑆)) +o ((𝐵 ·o 𝐺) ·o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅))))
53	49, 52	eqtrd 2264	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅))))
54	53	adantr 276	. . . 4 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ ((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))) = (((𝐵 ·o 𝐶) ·o (𝐺 ·o 𝑆)) +o ((𝐵 ·o 𝐷) ·o (𝐺 ·o 𝑅))))
55	39, 54	eqtr4d 2267	. . 3 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ ((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅))))
56		nnacl 6648	. . . . . 6 ⊢ (((𝐴 ·o 𝐺) ∈ ω ∧ (𝐵 ·o 𝐹) ∈ ω) → ((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ∈ ω)
57	8, 14, 56	syl2anc 411	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → ((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ∈ ω)
58		mulpiord 7537	. . . . . . . 8 ⊢ ((𝐵 ∈ N ∧ 𝐺 ∈ N) → (𝐵 ·N 𝐺) = (𝐵 ·o 𝐺))
59		mulclpi 7548	. . . . . . . 8 ⊢ ((𝐵 ∈ N ∧ 𝐺 ∈ N) → (𝐵 ·N 𝐺) ∈ N)
60	58, 59	eqeltrrd 2309	. . . . . . 7 ⊢ ((𝐵 ∈ N ∧ 𝐺 ∈ N) → (𝐵 ·o 𝐺) ∈ N)
61	60	ad2ant2l 508	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐹 ∈ ω ∧ 𝐺 ∈ N)) → (𝐵 ·o 𝐺) ∈ N)
62	61	ad2ant2r 509	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (𝐵 ·o 𝐺) ∈ N)
63		nnacl 6648	. . . . . 6 ⊢ (((𝐶 ·o 𝑆) ∈ ω ∧ (𝐷 ·o 𝑅) ∈ ω) → ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)) ∈ ω)
64	44, 47, 63	syl2anc 411	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)) ∈ ω)
65		mulpiord 7537	. . . . . . . 8 ⊢ ((𝐷 ∈ N ∧ 𝑆 ∈ N) → (𝐷 ·N 𝑆) = (𝐷 ·o 𝑆))
66		mulclpi 7548	. . . . . . . 8 ⊢ ((𝐷 ∈ N ∧ 𝑆 ∈ N) → (𝐷 ·N 𝑆) ∈ N)
67	65, 66	eqeltrrd 2309	. . . . . . 7 ⊢ ((𝐷 ∈ N ∧ 𝑆 ∈ N) → (𝐷 ·o 𝑆) ∈ N)
68	67	ad2ant2l 508	. . . . . 6 ⊢ (((𝐶 ∈ ω ∧ 𝐷 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N)) → (𝐷 ·o 𝑆) ∈ N)
69	68	ad2ant2l 508	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (𝐷 ·o 𝑆) ∈ N)
70		enq0breq 7656	. . . . 5 ⊢ (((((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ∈ ω ∧ (𝐵 ·o 𝐺) ∈ N) ∧ (((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)) ∈ ω ∧ (𝐷 ·o 𝑆) ∈ N)) → (⟨((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)⟩ ~Q0 ⟨((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)), (𝐷 ·o 𝑆)⟩ ↔ (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)))))
71	57, 62, 64, 69, 70	syl22anc 1274	. . . 4 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (⟨((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)⟩ ~Q0 ⟨((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)), (𝐷 ·o 𝑆)⟩ ↔ (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)))))
72	71	adantr 276	. . 3 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ ((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → (⟨((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)⟩ ~Q0 ⟨((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)), (𝐷 ·o 𝑆)⟩ ↔ (((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)) ·o (𝐷 ·o 𝑆)) = ((𝐵 ·o 𝐺) ·o ((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)))))
73	55, 72	mpbird 167	. 2 ⊢ (((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) ∧ ((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅))) → ⟨((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)⟩ ~Q0 ⟨((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)), (𝐷 ·o 𝑆)⟩)
74	73	ex 115	1 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) ∧ ((𝐹 ∈ ω ∧ 𝐺 ∈ N) ∧ (𝑅 ∈ ω ∧ 𝑆 ∈ N))) → (((𝐴 ·o 𝐷) = (𝐵 ·o 𝐶) ∧ (𝐹 ·o 𝑆) = (𝐺 ·o 𝑅)) → ⟨((𝐴 ·o 𝐺) +o (𝐵 ·o 𝐹)), (𝐵 ·o 𝐺)⟩ ~Q0 ⟨((𝐶 ·o 𝑆) +o (𝐷 ·o 𝑅)), (𝐷 ·o 𝑆)⟩))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ⟨cop 3672   class class class wbr 4088  ωcom 4688  (class class class)co 6018   +_o coa 6579   ·_o comu 6580  Ncnpi 7492   ·_N cmi 7494   ~_Q0 ceq0 7506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-oadd 6586  df-omul 6587  df-ni 7524  df-mi 7526  df-enq0 7644

This theorem is referenced by:  addnq0mo  7667