Theorem mulcmpblnrlem 10983

Description: Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
mulcmpblnrlem	⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))))

Proof of Theorem mulcmpblnrlem

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

Step	Hyp	Ref	Expression
1		oveq1 7360	. . . . . . . . 9 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐵 +P 𝐶) ·P 𝐹))
2		distrpr 10941	. . . . . . . . . 10 ⊢ (𝐹 ·P (𝐴 +P 𝐷)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷))
3		mulcompr 10936	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐷) ·P 𝐹) = (𝐹 ·P (𝐴 +P 𝐷))
4		mulcompr 10936	. . . . . . . . . . 11 ⊢ (𝐴 ·P 𝐹) = (𝐹 ·P 𝐴)
5		mulcompr 10936	. . . . . . . . . . 11 ⊢ (𝐷 ·P 𝐹) = (𝐹 ·P 𝐷)
6	4, 5	oveq12i 7365	. . . . . . . . . 10 ⊢ ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷))
7	2, 3, 6	3eqtr4i 2762	. . . . . . . . 9 ⊢ ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹))
8		distrpr 10941	. . . . . . . . . 10 ⊢ (𝐹 ·P (𝐵 +P 𝐶)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶))
9		mulcompr 10936	. . . . . . . . . 10 ⊢ ((𝐵 +P 𝐶) ·P 𝐹) = (𝐹 ·P (𝐵 +P 𝐶))
10		mulcompr 10936	. . . . . . . . . . 11 ⊢ (𝐵 ·P 𝐹) = (𝐹 ·P 𝐵)
11		mulcompr 10936	. . . . . . . . . . 11 ⊢ (𝐶 ·P 𝐹) = (𝐹 ·P 𝐶)
12	10, 11	oveq12i 7365	. . . . . . . . . 10 ⊢ ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶))
13	8, 9, 12	3eqtr4i 2762	. . . . . . . . 9 ⊢ ((𝐵 +P 𝐶) ·P 𝐹) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹))
14	1, 7, 13	3eqtr3g 2787	. . . . . . . 8 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)))
15	14	oveq1d 7368	. . . . . . 7 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)))
16		addasspr 10935	. . . . . . . 8 ⊢ (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)))
17		oveq2 7361	. . . . . . . . . 10 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐶 ·P (𝐹 +P 𝑆)) = (𝐶 ·P (𝐺 +P 𝑅)))
18		distrpr 10941	. . . . . . . . . 10 ⊢ (𝐶 ·P (𝐹 +P 𝑆)) = ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))
19		distrpr 10941	. . . . . . . . . 10 ⊢ (𝐶 ·P (𝐺 +P 𝑅)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))
20	17, 18, 19	3eqtr3g 2787	. . . . . . . . 9 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅)))
21	20	oveq2d 7369	. . . . . . . 8 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
22	16, 21	eqtrid 2776	. . . . . . 7 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
23	15, 22	sylan9eq 2784	. . . . . 6 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
24		ovex 7386	. . . . . . 7 ⊢ (𝐴 ·P 𝐹) ∈ V
25		ovex 7386	. . . . . . 7 ⊢ (𝐷 ·P 𝐹) ∈ V
26		ovex 7386	. . . . . . 7 ⊢ (𝐶 ·P 𝑆) ∈ V
27		addcompr 10934	. . . . . . 7 ⊢ (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
28		addasspr 10935	. . . . . . 7 ⊢ ((𝑥 +P 𝑦) +P 𝑧) = (𝑥 +P (𝑦 +P 𝑧))
29	24, 25, 26, 27, 28	caov32 7580	. . . . . 6 ⊢ (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))
30		ovex 7386	. . . . . . 7 ⊢ (𝐵 ·P 𝐹) ∈ V
31		ovex 7386	. . . . . . 7 ⊢ (𝐶 ·P 𝐺) ∈ V
32		ovex 7386	. . . . . . 7 ⊢ (𝐶 ·P 𝑅) ∈ V
33	30, 31, 32, 27, 28	caov12 7581	. . . . . 6 ⊢ ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))) = ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))
34	23, 29, 33	3eqtr3g 2787	. . . . 5 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹)) = ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
35	34	oveq2d 7369	. . . 4 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
36		oveq2 7361	. . . . . . . . . . 11 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐷 ·P (𝐹 +P 𝑆)) = (𝐷 ·P (𝐺 +P 𝑅)))
37		distrpr 10941	. . . . . . . . . . 11 ⊢ (𝐷 ·P (𝐹 +P 𝑆)) = ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))
38		distrpr 10941	. . . . . . . . . . 11 ⊢ (𝐷 ·P (𝐺 +P 𝑅)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))
39	36, 37, 38	3eqtr3g 2787	. . . . . . . . . 10 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
40	39	oveq2d 7369	. . . . . . . . 9 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))))
41		addasspr 10935	. . . . . . . . 9 ⊢ (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
42	40, 41	eqtr4di 2782	. . . . . . . 8 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)))
43		oveq1 7360	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐵 +P 𝐶) ·P 𝐺))
44		distrpr 10941	. . . . . . . . . . 11 ⊢ (𝐺 ·P (𝐴 +P 𝐷)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷))
45		mulcompr 10936	. . . . . . . . . . 11 ⊢ ((𝐴 +P 𝐷) ·P 𝐺) = (𝐺 ·P (𝐴 +P 𝐷))
46		mulcompr 10936	. . . . . . . . . . . 12 ⊢ (𝐴 ·P 𝐺) = (𝐺 ·P 𝐴)
47		mulcompr 10936	. . . . . . . . . . . 12 ⊢ (𝐷 ·P 𝐺) = (𝐺 ·P 𝐷)
48	46, 47	oveq12i 7365	. . . . . . . . . . 11 ⊢ ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷))
49	44, 45, 48	3eqtr4i 2762	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺))
50		distrpr 10941	. . . . . . . . . . 11 ⊢ (𝐺 ·P (𝐵 +P 𝐶)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶))
51		mulcompr 10936	. . . . . . . . . . 11 ⊢ ((𝐵 +P 𝐶) ·P 𝐺) = (𝐺 ·P (𝐵 +P 𝐶))
52		mulcompr 10936	. . . . . . . . . . . 12 ⊢ (𝐵 ·P 𝐺) = (𝐺 ·P 𝐵)
53		mulcompr 10936	. . . . . . . . . . . 12 ⊢ (𝐶 ·P 𝐺) = (𝐺 ·P 𝐶)
54	52, 53	oveq12i 7365	. . . . . . . . . . 11 ⊢ ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶))
55	50, 51, 54	3eqtr4i 2762	. . . . . . . . . 10 ⊢ ((𝐵 +P 𝐶) ·P 𝐺) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺))
56	43, 49, 55	3eqtr3g 2787	. . . . . . . . 9 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)))
57	56	oveq1d 7368	. . . . . . . 8 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)))
58	42, 57	sylan9eqr 2786	. . . . . . 7 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)))
59		ovex 7386	. . . . . . . 8 ⊢ (𝐴 ·P 𝐺) ∈ V
60		ovex 7386	. . . . . . . 8 ⊢ (𝐷 ·P 𝑆) ∈ V
61	59, 25, 60, 27, 28	caov12 7581	. . . . . . 7 ⊢ ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = ((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)))
62		ovex 7386	. . . . . . . 8 ⊢ (𝐵 ·P 𝐺) ∈ V
63		ovex 7386	. . . . . . . 8 ⊢ (𝐷 ·P 𝑅) ∈ V
64	62, 31, 63, 27, 28	caov32 7580	. . . . . . 7 ⊢ (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺))
65	58, 61, 64	3eqtr3g 2787	. . . . . 6 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)))
66	65	oveq1d 7368	. . . . 5 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = ((((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
67		addasspr 10935	. . . . 5 ⊢ ((((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
68	66, 67	eqtrdi 2780	. . . 4 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
69	35, 68	eqtr4d 2767	. . 3 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
70		ovex 7386	. . . 4 ⊢ ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) ∈ V
71		ovex 7386	. . . 4 ⊢ ((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) ∈ V
72	70, 71, 25, 27, 28	caov13 7583	. . 3 ⊢ (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅))))
73		addasspr 10935	. . 3 ⊢ (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
74	69, 72, 73	3eqtr3g 2787	. 2 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
75	24, 26, 62, 27, 28, 63	caov4 7584	. . 3 ⊢ (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅))) = (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))
76	75	oveq2i 7364	. 2 ⊢ ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅))))
77	59, 60, 30, 27, 28, 32	caov42 7586	. . 3 ⊢ (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))
78	77	oveq2i 7364	. 2 ⊢ ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆))))
79	74, 76, 78	3eqtr3g 2787	1 ⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  (class class class)co 7353   +_P cpp 10774   ·_P cmp 10775

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 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-er 8632  df-ni 10785  df-pli 10786  df-mi 10787  df-lti 10788  df-plpq 10821  df-mpq 10822  df-ltpq 10823  df-enq 10824  df-nq 10825  df-erq 10826  df-plq 10827  df-mq 10828  df-1nq 10829  df-rq 10830  df-ltnq 10831  df-np 10894  df-plp 10896  df-mp 10897

This theorem is referenced by:  mulcmpblnr  10984