Theorem ltmprr 7726

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 7722	. . . . 5 ⊢ (𝐶 ∈ P → ∃𝑦 ∈ P (𝐶 ·P 𝑦) = 1P)
2	1	3ad2ant3 1022	. . . 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 7697	. . . . 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 1011	. . . . . 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 7656	. . . . . . 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 7681	. . . . . 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 5941	. . . . . 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 1013	. . . . . . . . 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 7656	. . . . . . . . 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 7672	. . . . . . . 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 1249	. . . . . . 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 7665	. . . . . . . . 9 ⊢ ((𝑦 ∈ P ∧ 𝐶 ∈ P ∧ 𝐴 ∈ P) → ((𝑦 ·P 𝐶) ·P 𝐴) = (𝑦 ·P (𝐶 ·P 𝐴)))
22	8, 16, 7, 21	syl3anc 1249	. . . . . . . 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 5940	. . . . . . 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 7664	. . . . . . . . . . . 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 2229	. . . . . . . . . 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 5940	. . . . . . . . 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 7638	. . . . . . . . . . . 12 ⊢ 1P ∈ P
30		mulcomprg 7664	. . . . . . . . . . . 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 7676	. . . . . . . . . . 11 ⊢ (𝐴 ∈ P → (𝐴 ·P 1P) = 𝐴)
33	31, 32	eqtr3d 2231	. . . . . . . . . 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 2229	. . . . . . . 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 5940	. . . . . . 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 2235	. . . . . 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 5940	. . . . . . 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 1012	. . . . . . . 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 7665	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) → ((𝑦 ·P 𝐶) ·P 𝐵) = (𝑦 ·P (𝐶 ·P 𝐵)))
41	8, 16, 39, 40	syl3anc 1249	. . . . . . 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 7664	. . . . . . . . . 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 7676	. . . . . . . . 9 ⊢ (𝐵 ∈ P → (𝐵 ·P 1P) = 𝐵)
45	43, 44	eqtr3d 2231	. . . . . . . 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 2237	. . . . . 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 2237	. . . . 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 4060	. . . 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 2619	. . 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 2619	. 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 980   = wceq 1364   ∈ wcel 2167  ∃wrex 2476   class class class wbr 4034  (class class class)co 5925  Pcnp 7375  1_Pc1p 7376   +_P cpp 7377   ·_P cmp 7378  <_P cltp 7379

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-i1p 7551  df-iplp 7552  df-imp 7553  df-iltp 7554

This theorem is referenced by:  mulextsr1lem  7864