Theorem ltmprr 7973

Description: Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)

Assertion

Ref	Expression
ltmprr	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))

Proof of Theorem ltmprr

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

Step	Hyp	Ref	Expression
1		recexpr 7969	. . . . 5 ⊢ (𝐶 ∈ P → ∃𝑦 ∈ P (𝐶 ·P 𝑦) = 1P)
2	1	3ad2ant3 1047	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ∃𝑦 ∈ P (𝐶 ·P 𝑦) = 1P)
3	2	adantr 276	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → ∃𝑦 ∈ P (𝐶 ·P 𝑦) = 1P)
4		ltexpri 7944	. . . . 5 ⊢ ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → ∃𝑥 ∈ P ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
5	4	ad2antlr 489	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) → ∃𝑥 ∈ P ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
6		simplll 535	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P))
7	6	simp1d 1036	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐴 ∈ P)
8		simplrl 537	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝑦 ∈ P)
9		simprl 531	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝑥 ∈ P)
10		mulclpr 7903	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑥 ∈ P) → (𝑦 ·P 𝑥) ∈ P)
11	8, 9, 10	syl2anc 411	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P 𝑥) ∈ P)
12		ltaddpr 7928	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ (𝑦 ·P 𝑥) ∈ P) → 𝐴<P (𝐴 +P (𝑦 ·P 𝑥)))
13	7, 11, 12	syl2anc 411	. . . . 5 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐴<P (𝐴 +P (𝑦 ·P 𝑥)))
14		simprr 533	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
15	14	oveq2d 6074	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P ((𝐶 ·P 𝐴) +P 𝑥)) = (𝑦 ·P (𝐶 ·P 𝐵)))
16	6	simp3d 1038	. . . . . . . . 9 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐶 ∈ P)
17		mulclpr 7903	. . . . . . . . 9 ⊢ ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝐶 ·P 𝐴) ∈ P)
18	16, 7, 17	syl2anc 411	. . . . . . . 8 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝐶 ·P 𝐴) ∈ P)
19		distrprg 7919	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ (𝐶 ·P 𝐴) ∈ P ∧ 𝑥 ∈ P) → (𝑦 ·P ((𝐶 ·P 𝐴) +P 𝑥)) = ((𝑦 ·P (𝐶 ·P 𝐴)) +P (𝑦 ·P 𝑥)))
20	8, 18, 9, 19	syl3anc 1274	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P ((𝐶 ·P 𝐴) +P 𝑥)) = ((𝑦 ·P (𝐶 ·P 𝐴)) +P (𝑦 ·P 𝑥)))
21		mulassprg 7912	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ 𝐶 ∈ P ∧ 𝐴 ∈ P) → ((𝑦 ·P 𝐶) ·P 𝐴) = (𝑦 ·P (𝐶 ·P 𝐴)))
22	8, 16, 7, 21	syl3anc 1274	. . . . . . . 8 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐴) = (𝑦 ·P (𝐶 ·P 𝐴)))
23	22	oveq1d 6073	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (((𝑦 ·P 𝐶) ·P 𝐴) +P (𝑦 ·P 𝑥)) = ((𝑦 ·P (𝐶 ·P 𝐴)) +P (𝑦 ·P 𝑥)))
24		mulcomprg 7911	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ P ∧ 𝐶 ∈ P) → (𝑦 ·P 𝐶) = (𝐶 ·P 𝑦))
25	8, 16, 24	syl2anc 411	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P 𝐶) = (𝐶 ·P 𝑦))
26		simplrr 538	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝐶 ·P 𝑦) = 1P)
27	25, 26	eqtrd 2267	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P 𝐶) = 1P)
28	27	oveq1d 6073	. . . . . . . . 9 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐴) = (1P ·P 𝐴))
29		1pr 7885	. . . . . . . . . . . 12 ⊢ 1P ∈ P
30		mulcomprg 7911	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 ·P 1P) = (1P ·P 𝐴))
31	29, 30	mpan2 425	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → (𝐴 ·P 1P) = (1P ·P 𝐴))
32		1idpr 7923	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → (𝐴 ·P 1P) = 𝐴)
33	31, 32	eqtr3d 2269	. . . . . . . . . 10 ⊢ (𝐴 ∈ P → (1P ·P 𝐴) = 𝐴)
34	7, 33	syl 14	. . . . . . . . 9 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (1P ·P 𝐴) = 𝐴)
35	28, 34	eqtrd 2267	. . . . . . . 8 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐴) = 𝐴)
36	35	oveq1d 6073	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (((𝑦 ·P 𝐶) ·P 𝐴) +P (𝑦 ·P 𝑥)) = (𝐴 +P (𝑦 ·P 𝑥)))
37	20, 23, 36	3eqtr2d 2273	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P ((𝐶 ·P 𝐴) +P 𝑥)) = (𝐴 +P (𝑦 ·P 𝑥)))
38	27	oveq1d 6073	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐵) = (1P ·P 𝐵))
39	6	simp2d 1037	. . . . . . . 8 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐵 ∈ P)
40		mulassprg 7912	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) → ((𝑦 ·P 𝐶) ·P 𝐵) = (𝑦 ·P (𝐶 ·P 𝐵)))
41	8, 16, 39, 40	syl3anc 1274	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → ((𝑦 ·P 𝐶) ·P 𝐵) = (𝑦 ·P (𝐶 ·P 𝐵)))
42		mulcomprg 7911	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → (𝐵 ·P 1P) = (1P ·P 𝐵))
43	29, 42	mpan2 425	. . . . . . . . 9 ⊢ (𝐵 ∈ P → (𝐵 ·P 1P) = (1P ·P 𝐵))
44		1idpr 7923	. . . . . . . . 9 ⊢ (𝐵 ∈ P → (𝐵 ·P 1P) = 𝐵)
45	43, 44	eqtr3d 2269	. . . . . . . 8 ⊢ (𝐵 ∈ P → (1P ·P 𝐵) = 𝐵)
46	39, 45	syl 14	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (1P ·P 𝐵) = 𝐵)
47	38, 41, 46	3eqtr3d 2275	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝑦 ·P (𝐶 ·P 𝐵)) = 𝐵)
48	15, 37, 47	3eqtr3d 2275	. . . . 5 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → (𝐴 +P (𝑦 ·P 𝑥)) = 𝐵)
49	13, 48	breqtrd 4140	. . . 4 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) ∧ (𝑥 ∈ P ∧ ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))) → 𝐴<P 𝐵)
50	5, 49	rexlimddv 2667	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) ∧ (𝑦 ∈ P ∧ (𝐶 ·P 𝑦) = 1P)) → 𝐴<P 𝐵)
51	3, 50	rexlimddv 2667	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵)) → 𝐴<P 𝐵)
52	51	ex 115	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐶 ·P 𝐴)<P (𝐶 ·P 𝐵) → 𝐴<P 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∃wrex 2523   class class class wbr 4114  (class class class)co 6058  Pcnp 7622  1_Pc1p 7623   +_P cpp 7624   ·_P cmp 7625  <_P cltp 7626

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-i1p 7798  df-iplp 7799  df-imp 7800  df-iltp 7801

This theorem is referenced by:  mulextsr1lem  8111