Theorem pgpfac1lem1 20296

Description: Lemma for pgpfac1 20302. (Contributed by Mario Carneiro, 27-Apr-2016.)

Hypotheses

Ref	Expression
pgpfac1.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
pgpfac1.s	⊢ 𝑆 = (𝐾‘{𝐴})
pgpfac1.b	⊢ 𝐵 = (Base‘𝐺)
pgpfac1.o	⊢ 𝑂 = (od‘𝐺)
pgpfac1.e	⊢ 𝐸 = (gEx‘𝐺)
pgpfac1.z	⊢ 0 = (0g‘𝐺)
pgpfac1.l	⊢ ⊕ = (LSSum‘𝐺)
pgpfac1.p	⊢ (𝜑 → 𝑃 pGrp 𝐺)
pgpfac1.g	⊢ (𝜑 → 𝐺 ∈ Abel)
pgpfac1.n	⊢ (𝜑 → 𝐵 ∈ Fin)
pgpfac1.oe	⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)
pgpfac1.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
pgpfac1.au	⊢ (𝜑 → 𝐴 ∈ 𝑈)
pgpfac1.w	⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
pgpfac1.i	⊢ (𝜑 → (𝑆 ∩ 𝑊) = { 0 })
pgpfac1.ss	⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
pgpfac1.2	⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))

Assertion

Ref	Expression
pgpfac1lem1	⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈)

Distinct variable groups:   𝑤,𝐴   𝑤, ⊕   𝑤,𝑃   𝑤,𝐺   𝑤,𝑈   𝑤,𝐶   𝑤,𝑆   𝑤,𝑊   𝜑,𝑤   𝑤,𝐾

Allowed substitution hints:   𝐵(𝑤)   𝐸(𝑤)   𝑂(𝑤)   0 (𝑤)

Proof of Theorem pgpfac1lem1

Step	Hyp	Ref	Expression
1		pgpfac1.ss	. . . 4 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
2	1	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
3		pgpfac1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Abel)
4		ablgrp 20005	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
5		pgpfac1.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
6	5	subgacs 19377	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
7		acsmre 17832	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘𝐵) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
8	3, 4, 6, 7	4syl 20	. . . . 5 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
9	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
10		eldifi 4067	. . . . . 6 ⊢ (𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)) → 𝐶 ∈ 𝑈)
11	10	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ 𝑈)
12	11	snssd 4743	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ 𝑈)
13		pgpfac1.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
14	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝑈 ∈ (SubGrp‘𝐺))
15		pgpfac1.k	. . . . 5 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
16	15	mrcsscl 17800	. . . 4 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐶} ⊆ 𝑈 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐶}) ⊆ 𝑈)
17	9, 12, 14, 16	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐾‘{𝐶}) ⊆ 𝑈)
18		pgpfac1.s	. . . . . . 7 ⊢ 𝑆 = (𝐾‘{𝐴})
19	5	subgss 19343	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
20	13, 19	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
21		pgpfac1.au	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
22	20, 21	sseldd 3909	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
23	15	mrcsncl 17792	. . . . . . . 8 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
24	8, 22, 23	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
25	18, 24	eqeltrid 2864	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
26		pgpfac1.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
27		pgpfac1.l	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝐺)
28	27	lsmsubg2 20079	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
29	3, 25, 26, 28	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
30	29	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
31	20	sselda 3908	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑈) → 𝐶 ∈ 𝐵)
32	10, 31	sylan2 605	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ 𝐵)
33	15	mrcsncl 17792	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺))
34	9, 32, 33	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺))
35	27	lsmlub 19884	. . . 4 ⊢ (((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (((𝑆 ⊕ 𝑊) ⊆ 𝑈 ∧ (𝐾‘{𝐶}) ⊆ 𝑈) ↔ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈))
36	30, 34, 14, 35	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (((𝑆 ⊕ 𝑊) ⊆ 𝑈 ∧ (𝐾‘{𝐶}) ⊆ 𝑈) ↔ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈))
37	2, 17, 36	mpbi2and 725	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈)
38	27	lsmub1 19877	. . . . . 6 ⊢ (((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
39	30, 34, 38	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
40	27	lsmub2 19878	. . . . . . 7 ⊢ (((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐶}) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
41	30, 34, 40	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐾‘{𝐶}) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
42	32	snssd 4743	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ 𝐵)
43	9, 15, 42	mrcssidd 17805	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ (𝐾‘{𝐶}))
44		snssg 4740	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐵 → (𝐶 ∈ (𝐾‘{𝐶}) ↔ {𝐶} ⊆ (𝐾‘{𝐶})))
45	32, 44	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐶 ∈ (𝐾‘{𝐶}) ↔ {𝐶} ⊆ (𝐾‘{𝐶})))
46	43, 45	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ (𝐾‘{𝐶}))
47	41, 46	sseldd 3909	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
48		eldifn 4068	. . . . . 6 ⊢ (𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)) → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
49	48	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
50	39, 47, 49	ssnelpssd 4043	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
51	27	lsmub1 19877	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑊))
52	25, 26, 51	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑊))
53	22	snssd 4743	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
54	8, 15, 53	mrcssidd 17805	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
55	54, 18	sseqtrrdi 3949	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
56		snssg 4740	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑈 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
57	21, 56	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
58	55, 57	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
59	52, 58	sseldd 3909	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝑆 ⊕ 𝑊))
60	59	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐴 ∈ (𝑆 ⊕ 𝑊))
61	39, 60	sseldd 3909	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
62		psseq1 4017	. . . . . . . 8 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (𝑤 ⊊ 𝑈 ↔ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈))
63		eleq2 2849	. . . . . . . 8 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
64	62, 63	anbi12d 644	. . . . . . 7 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → ((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) ↔ (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ∧ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))))
65		psseq2 4018	. . . . . . . 8 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → ((𝑆 ⊕ 𝑊) ⊊ 𝑤 ↔ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
66	65	notbid 321	. . . . . . 7 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤 ↔ ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
67	64, 66	imbi12d 347	. . . . . 6 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤) ↔ ((((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ∧ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))) → ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))))
68		pgpfac1.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))
69	68	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))
70	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐺 ∈ Abel)
71	27	lsmsubg2 20079	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺)) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ∈ (SubGrp‘𝐺))
72	70, 30, 34, 71	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ∈ (SubGrp‘𝐺))
73	67, 69, 72	rspcdva 3577	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ∧ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))) → ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
74	61, 73	mpan2d 707	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 → ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
75	50, 74	mt2d 137	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ¬ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈)
76		npss 4041	. . 3 ⊢ (¬ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ↔ (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈 → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈))
77	75, 76	sylib 221	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈 → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈))
78	37, 77	mpd 16	1 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ⊊ wpss 4013   ∖ cdif 4045   ∩ cin 4089  {csn 4584   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  Fincfn 8977  Basecbs 17393  0_gc0g 17616  Moorecmre 17758  mrClscmrc 17759  ACScacs 17761  Grpcgrp 19150  SubGrpcsubg 19336  odcod 19744  gExcgex 19745   pGrp cpgp 19746  LSSumclsm 19854  Abelcabl 20001

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-ress 17415  df-plusg 17447  df-0g 17618  df-mre 17762  df-mrc 17763  df-acs 17765  df-mgm 18822  df-sgrp 18914  df-mnd 18930  df-submnd 18985  df-grp 19153  df-minusg 19154  df-subg 19339  df-cntz 19537  df-lsm 19856  df-cmn 20002  df-abl 20003

This theorem is used by:  pgpfac1lem2  20297  pgpfac1lem3  20299