Theorem ltmprr 7632

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 7628	. . . . 5 ⊢ (𝐶 ∈ P → ∃𝑦 ∈ P (𝐶 ·P 𝑦) = 1P)
2	1	3ad2ant3 1020	. . . 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 7603	. . . . 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 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))) → (𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P))
7	6	simp1d 1009	. . . . . 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 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))) → 𝑦 ∈ P)
9		simprl 529	. . . . . . 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 7562	. . . . . . 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 7587	. . . . . 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 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))) → ((𝐶 ·P 𝐴) +P 𝑥) = (𝐶 ·P 𝐵))
15	14	oveq2d 5885	. . . . . 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 1011	. . . . . . . . 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 7562	. . . . . . . . 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 7578	. . . . . . . 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 1238	. . . . . . 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 7571	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ 𝐶 ∈ P ∧ 𝐴 ∈ P) → ((𝑦 ·P 𝐶) ·P 𝐴) = (𝑦 ·P (𝐶 ·P 𝐴)))
22	8, 16, 7, 21	syl3anc 1238	. . . . . . . 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 5884	. . . . . . 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 7570	. . . . . . . . . . . 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 536	. . . . . . . . . . 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 2210	. . . . . . . . . 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 5884	. . . . . . . . 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 7544	. . . . . . . . . . . 12 ⊢ 1P ∈ P
30		mulcomprg 7570	. . . . . . . . . . . 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 7582	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → (𝐴 ·P 1P) = 𝐴)
33	31, 32	eqtr3d 2212	. . . . . . . . . 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 2210	. . . . . . . 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 5884	. . . . . . 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 2216	. . . . . 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 5884	. . . . . . 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 1010	. . . . . . . 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 7571	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) → ((𝑦 ·P 𝐶) ·P 𝐵) = (𝑦 ·P (𝐶 ·P 𝐵)))
41	8, 16, 39, 40	syl3anc 1238	. . . . . . 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 7570	. . . . . . . . . 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 7582	. . . . . . . . 9 ⊢ (𝐵 ∈ P → (𝐵 ·P 1P) = 𝐵)
45	43, 44	eqtr3d 2212	. . . . . . . 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 2218	. . . . . 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 2218	. . . . 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 4026	. . . 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 2599	. . 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 2599	. 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 978   = wceq 1353   ∈ wcel 2148  ∃wrex 2456   class class class wbr 4000  (class class class)co 5869  Pcnp 7281  1_Pc1p 7282   +_P cpp 7283   ·_P cmp 7284  <_P cltp 7285

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-imp 7459  df-iltp 7460

This theorem is referenced by:  mulextsr1lem  7770