Theorem mulcmpblnrlem 10229

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 6931	. . . . . . . . 9 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐵 +P 𝐶) ·P 𝐹))
2		distrpr 10187	. . . . . . . . . 10 ⊢ (𝐹 ·P (𝐴 +P 𝐷)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷))
3		mulcompr 10182	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐷) ·P 𝐹) = (𝐹 ·P (𝐴 +P 𝐷))
4		mulcompr 10182	. . . . . . . . . . 11 ⊢ (𝐴 ·P 𝐹) = (𝐹 ·P 𝐴)
5		mulcompr 10182	. . . . . . . . . . 11 ⊢ (𝐷 ·P 𝐹) = (𝐹 ·P 𝐷)
6	4, 5	oveq12i 6936	. . . . . . . . . 10 ⊢ ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷))
7	2, 3, 6	3eqtr4i 2812	. . . . . . . . 9 ⊢ ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹))
8		distrpr 10187	. . . . . . . . . 10 ⊢ (𝐹 ·P (𝐵 +P 𝐶)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶))
9		mulcompr 10182	. . . . . . . . . 10 ⊢ ((𝐵 +P 𝐶) ·P 𝐹) = (𝐹 ·P (𝐵 +P 𝐶))
10		mulcompr 10182	. . . . . . . . . . 11 ⊢ (𝐵 ·P 𝐹) = (𝐹 ·P 𝐵)
11		mulcompr 10182	. . . . . . . . . . 11 ⊢ (𝐶 ·P 𝐹) = (𝐹 ·P 𝐶)
12	10, 11	oveq12i 6936	. . . . . . . . . 10 ⊢ ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶))
13	8, 9, 12	3eqtr4i 2812	. . . . . . . . 9 ⊢ ((𝐵 +P 𝐶) ·P 𝐹) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹))
14	1, 7, 13	3eqtr3g 2837	. . . . . . . 8 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)))
15	14	oveq1d 6939	. . . . . . 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 10181	. . . . . . . 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 6932	. . . . . . . . . 10 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐶 ·P (𝐹 +P 𝑆)) = (𝐶 ·P (𝐺 +P 𝑅)))
18		distrpr 10187	. . . . . . . . . 10 ⊢ (𝐶 ·P (𝐹 +P 𝑆)) = ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))
19		distrpr 10187	. . . . . . . . . 10 ⊢ (𝐶 ·P (𝐺 +P 𝑅)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))
20	17, 18, 19	3eqtr3g 2837	. . . . . . . . 9 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅)))
21	20	oveq2d 6940	. . . . . . . 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	syl5eq 2826	. . . . . . 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 2834	. . . . . 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 6956	. . . . . . 7 ⊢ (𝐴 ·P 𝐹) ∈ V
25		ovex 6956	. . . . . . 7 ⊢ (𝐷 ·P 𝐹) ∈ V
26		ovex 6956	. . . . . . 7 ⊢ (𝐶 ·P 𝑆) ∈ V
27		addcompr 10180	. . . . . . 7 ⊢ (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
28		addasspr 10181	. . . . . . 7 ⊢ ((𝑥 +P 𝑦) +P 𝑧) = (𝑥 +P (𝑦 +P 𝑧))
29	24, 25, 26, 27, 28	caov32 7140	. . . . . 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 6956	. . . . . . 7 ⊢ (𝐵 ·P 𝐹) ∈ V
31		ovex 6956	. . . . . . 7 ⊢ (𝐶 ·P 𝐺) ∈ V
32		ovex 6956	. . . . . . 7 ⊢ (𝐶 ·P 𝑅) ∈ V
33	30, 31, 32, 27, 28	caov12 7141	. . . . . 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 2837	. . . . 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 6940	. . . 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 6932	. . . . . . . . . . 11 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐷 ·P (𝐹 +P 𝑆)) = (𝐷 ·P (𝐺 +P 𝑅)))
37		distrpr 10187	. . . . . . . . . . 11 ⊢ (𝐷 ·P (𝐹 +P 𝑆)) = ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))
38		distrpr 10187	. . . . . . . . . . 11 ⊢ (𝐷 ·P (𝐺 +P 𝑅)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))
39	36, 37, 38	3eqtr3g 2837	. . . . . . . . . 10 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
40	39	oveq2d 6940	. . . . . . . . 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 10181	. . . . . . . . 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	syl6eqr 2832	. . . . . . . 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 6931	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐵 +P 𝐶) ·P 𝐺))
44		distrpr 10187	. . . . . . . . . . 11 ⊢ (𝐺 ·P (𝐴 +P 𝐷)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷))
45		mulcompr 10182	. . . . . . . . . . 11 ⊢ ((𝐴 +P 𝐷) ·P 𝐺) = (𝐺 ·P (𝐴 +P 𝐷))
46		mulcompr 10182	. . . . . . . . . . . 12 ⊢ (𝐴 ·P 𝐺) = (𝐺 ·P 𝐴)
47		mulcompr 10182	. . . . . . . . . . . 12 ⊢ (𝐷 ·P 𝐺) = (𝐺 ·P 𝐷)
48	46, 47	oveq12i 6936	. . . . . . . . . . 11 ⊢ ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷))
49	44, 45, 48	3eqtr4i 2812	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺))
50		distrpr 10187	. . . . . . . . . . 11 ⊢ (𝐺 ·P (𝐵 +P 𝐶)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶))
51		mulcompr 10182	. . . . . . . . . . 11 ⊢ ((𝐵 +P 𝐶) ·P 𝐺) = (𝐺 ·P (𝐵 +P 𝐶))
52		mulcompr 10182	. . . . . . . . . . . 12 ⊢ (𝐵 ·P 𝐺) = (𝐺 ·P 𝐵)
53		mulcompr 10182	. . . . . . . . . . . 12 ⊢ (𝐶 ·P 𝐺) = (𝐺 ·P 𝐶)
54	52, 53	oveq12i 6936	. . . . . . . . . . 11 ⊢ ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶))
55	50, 51, 54	3eqtr4i 2812	. . . . . . . . . 10 ⊢ ((𝐵 +P 𝐶) ·P 𝐺) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺))
56	43, 49, 55	3eqtr3g 2837	. . . . . . . . 9 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)))
57	56	oveq1d 6939	. . . . . . . 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 2836	. . . . . . 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 6956	. . . . . . . 8 ⊢ (𝐴 ·P 𝐺) ∈ V
60		ovex 6956	. . . . . . . 8 ⊢ (𝐷 ·P 𝑆) ∈ V
61	59, 25, 60, 27, 28	caov12 7141	. . . . . . 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 6956	. . . . . . . 8 ⊢ (𝐵 ·P 𝐺) ∈ V
63		ovex 6956	. . . . . . . 8 ⊢ (𝐷 ·P 𝑅) ∈ V
64	62, 31, 63, 27, 28	caov32 7140	. . . . . . 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 2837	. . . . . 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 6939	. . . . 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 10181	. . . . 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	syl6eq 2830	. . . 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 2817	. . 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 6956	. . . 4 ⊢ ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) ∈ V
71		ovex 6956	. . . 4 ⊢ ((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) ∈ V
72	70, 71, 25, 27, 28	caov13 7143	. . 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 10181	. . 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 2837	. 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 7144	. . 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 6935	. 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 7146	. . 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 6935	. 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 2837	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 386   = wceq 1601  (class class class)co 6924   +_P cpp 10020   ·_P cmp 10021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-omul 7850  df-er 8028  df-ni 10031  df-pli 10032  df-mi 10033  df-lti 10034  df-plpq 10067  df-mpq 10068  df-ltpq 10069  df-enq 10070  df-nq 10071  df-erq 10072  df-plq 10073  df-mq 10074  df-1nq 10075  df-rq 10076  df-ltnq 10077  df-np 10140  df-plp 10142  df-mp 10143

This theorem is referenced by:  mulcmpblnr  10230