Theorem mulcmpblnrlem 11018

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 7392	. . . . . . . . 9 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐵 +P 𝐶) ·P 𝐹))
2		distrpr 10976	. . . . . . . . . 10 ⊢ (𝐹 ·P (𝐴 +P 𝐷)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷))
3		mulcompr 10971	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐷) ·P 𝐹) = (𝐹 ·P (𝐴 +P 𝐷))
4		mulcompr 10971	. . . . . . . . . . 11 ⊢ (𝐴 ·P 𝐹) = (𝐹 ·P 𝐴)
5		mulcompr 10971	. . . . . . . . . . 11 ⊢ (𝐷 ·P 𝐹) = (𝐹 ·P 𝐷)
6	4, 5	oveq12i 7397	. . . . . . . . . 10 ⊢ ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷))
7	2, 3, 6	3eqtr4i 2789	. . . . . . . . 9 ⊢ ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹))
8		distrpr 10976	. . . . . . . . . 10 ⊢ (𝐹 ·P (𝐵 +P 𝐶)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶))
9		mulcompr 10971	. . . . . . . . . 10 ⊢ ((𝐵 +P 𝐶) ·P 𝐹) = (𝐹 ·P (𝐵 +P 𝐶))
10		mulcompr 10971	. . . . . . . . . . 11 ⊢ (𝐵 ·P 𝐹) = (𝐹 ·P 𝐵)
11		mulcompr 10971	. . . . . . . . . . 11 ⊢ (𝐶 ·P 𝐹) = (𝐹 ·P 𝐶)
12	10, 11	oveq12i 7397	. . . . . . . . . 10 ⊢ ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶))
13	8, 9, 12	3eqtr4i 2789	. . . . . . . . 9 ⊢ ((𝐵 +P 𝐶) ·P 𝐹) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹))
14	1, 7, 13	3eqtr3g 2814	. . . . . . . 8 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)))
15	14	oveq1d 7400	. . . . . . 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 10970	. . . . . . . 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 7393	. . . . . . . . . 10 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐶 ·P (𝐹 +P 𝑆)) = (𝐶 ·P (𝐺 +P 𝑅)))
18		distrpr 10976	. . . . . . . . . 10 ⊢ (𝐶 ·P (𝐹 +P 𝑆)) = ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))
19		distrpr 10976	. . . . . . . . . 10 ⊢ (𝐶 ·P (𝐺 +P 𝑅)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))
20	17, 18, 19	3eqtr3g 2814	. . . . . . . . 9 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅)))
21	20	oveq2d 7401	. . . . . . . 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 2803	. . . . . . 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 2811	. . . . . 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 7418	. . . . . . 7 ⊢ (𝐴 ·P 𝐹) ∈ V
25		ovex 7418	. . . . . . 7 ⊢ (𝐷 ·P 𝐹) ∈ V
26		ovex 7418	. . . . . . 7 ⊢ (𝐶 ·P 𝑆) ∈ V
27		addcompr 10969	. . . . . . 7 ⊢ (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
28		addasspr 10970	. . . . . . 7 ⊢ ((𝑥 +P 𝑦) +P 𝑧) = (𝑥 +P (𝑦 +P 𝑧))
29	24, 25, 26, 27, 28	caov32 7612	. . . . . 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 7418	. . . . . . 7 ⊢ (𝐵 ·P 𝐹) ∈ V
31		ovex 7418	. . . . . . 7 ⊢ (𝐶 ·P 𝐺) ∈ V
32		ovex 7418	. . . . . . 7 ⊢ (𝐶 ·P 𝑅) ∈ V
33	30, 31, 32, 27, 28	caov12 7613	. . . . . 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 2814	. . . . 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 7401	. . . 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 7393	. . . . . . . . . . 11 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐷 ·P (𝐹 +P 𝑆)) = (𝐷 ·P (𝐺 +P 𝑅)))
37		distrpr 10976	. . . . . . . . . . 11 ⊢ (𝐷 ·P (𝐹 +P 𝑆)) = ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))
38		distrpr 10976	. . . . . . . . . . 11 ⊢ (𝐷 ·P (𝐺 +P 𝑅)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))
39	36, 37, 38	3eqtr3g 2814	. . . . . . . . . 10 ⊢ ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
40	39	oveq2d 7401	. . . . . . . . 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 10970	. . . . . . . . 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 2809	. . . . . . . 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 7392	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐵 +P 𝐶) ·P 𝐺))
44		distrpr 10976	. . . . . . . . . . 11 ⊢ (𝐺 ·P (𝐴 +P 𝐷)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷))
45		mulcompr 10971	. . . . . . . . . . 11 ⊢ ((𝐴 +P 𝐷) ·P 𝐺) = (𝐺 ·P (𝐴 +P 𝐷))
46		mulcompr 10971	. . . . . . . . . . . 12 ⊢ (𝐴 ·P 𝐺) = (𝐺 ·P 𝐴)
47		mulcompr 10971	. . . . . . . . . . . 12 ⊢ (𝐷 ·P 𝐺) = (𝐺 ·P 𝐷)
48	46, 47	oveq12i 7397	. . . . . . . . . . 11 ⊢ ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷))
49	44, 45, 48	3eqtr4i 2789	. . . . . . . . . 10 ⊢ ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺))
50		distrpr 10976	. . . . . . . . . . 11 ⊢ (𝐺 ·P (𝐵 +P 𝐶)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶))
51		mulcompr 10971	. . . . . . . . . . 11 ⊢ ((𝐵 +P 𝐶) ·P 𝐺) = (𝐺 ·P (𝐵 +P 𝐶))
52		mulcompr 10971	. . . . . . . . . . . 12 ⊢ (𝐵 ·P 𝐺) = (𝐺 ·P 𝐵)
53		mulcompr 10971	. . . . . . . . . . . 12 ⊢ (𝐶 ·P 𝐺) = (𝐺 ·P 𝐶)
54	52, 53	oveq12i 7397	. . . . . . . . . . 11 ⊢ ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶))
55	50, 51, 54	3eqtr4i 2789	. . . . . . . . . 10 ⊢ ((𝐵 +P 𝐶) ·P 𝐺) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺))
56	43, 49, 55	3eqtr3g 2814	. . . . . . . . 9 ⊢ ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)))
57	56	oveq1d 7400	. . . . . . . 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 2813	. . . . . . 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 7418	. . . . . . . 8 ⊢ (𝐴 ·P 𝐺) ∈ V
60		ovex 7418	. . . . . . . 8 ⊢ (𝐷 ·P 𝑆) ∈ V
61	59, 25, 60, 27, 28	caov12 7613	. . . . . . 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 7418	. . . . . . . 8 ⊢ (𝐵 ·P 𝐺) ∈ V
63		ovex 7418	. . . . . . . 8 ⊢ (𝐷 ·P 𝑅) ∈ V
64	62, 31, 63, 27, 28	caov32 7612	. . . . . . 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 2814	. . . . . 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 7400	. . . . 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 10970	. . . . 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 2807	. . . 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 2794	. . 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 7418	. . . 4 ⊢ ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) ∈ V
71		ovex 7418	. . . 4 ⊢ ((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) ∈ V
72	70, 71, 25, 27, 28	caov13 7615	. . 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 10970	. . 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 2814	. 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 7616	. . 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 7396	. 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 7618	. . 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 7396	. 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 2814	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 398   = wceq 1554  (class class class)co 7385   +_P cpp 10809   ·_P cmp 10810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-inf2 9586

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-1o 8425  df-oadd 8429  df-omul 8430  df-er 8666  df-ni 10820  df-pli 10821  df-mi 10822  df-lti 10823  df-plpq 10856  df-mpq 10857  df-ltpq 10858  df-enq 10859  df-nq 10860  df-erq 10861  df-plq 10862  df-mq 10863  df-1nq 10864  df-rq 10865  df-ltnq 10866  df-np 10929  df-plp 10931  df-mp 10932

This theorem is referenced by:  mulcmpblnr  11019