Theorem pgpfac1lem1 18296

Description: Lemma for pgpfac1 18302. (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 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊆ 𝑈)
3		pgpfac1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Abel)
4		ablgrp 18021	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
5		pgpfac1.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
6	5	subgacs 17452	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
7		acsmre 16136	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘𝐵) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
8	3, 4, 6, 7	4syl 19	. . . . 5 ⊢ (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (SubGrp‘𝐺) ∈ (Moore‘𝐵))
10		eldifi 3694	. . . . . 6 ⊢ (𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)) → 𝐶 ∈ 𝑈)
11	10	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ 𝑈)
12	11	snssd 4281	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ 𝑈)
13		pgpfac1.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝑈 ∈ (SubGrp‘𝐺))
15		pgpfac1.k	. . . . 5 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
16	15	mrcsscl 16103	. . . 4 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ {𝐶} ⊆ 𝑈 ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐶}) ⊆ 𝑈)
17	9, 12, 14, 16	syl3anc 1318	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐾‘{𝐶}) ⊆ 𝑈)
18		pgpfac1.s	. . . . . . 7 ⊢ 𝑆 = (𝐾‘{𝐴})
19	5	subgss 17418	. . . . . . . . . 10 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ 𝐵)
20	13, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
21		pgpfac1.au	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
22	20, 21	sseldd 3569	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
23	15	mrcsncl 16095	. . . . . . . 8 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐴 ∈ 𝐵) → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
24	8, 22, 23	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝐾‘{𝐴}) ∈ (SubGrp‘𝐺))
25	18, 24	syl5eqel 2692	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
26		pgpfac1.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
27		pgpfac1.l	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝐺)
28	27	lsmsubg2 18085	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
29	3, 25, 26, 28	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
30	29	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺))
31	20	sselda 3568	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑈) → 𝐶 ∈ 𝐵)
32	10, 31	sylan2 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ 𝐵)
33	15	mrcsncl 16095	. . . . 5 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺))
34	9, 32, 33	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺))
35	27	lsmlub 17901	. . . 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 1318	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (((𝑆 ⊕ 𝑊) ⊆ 𝑈 ∧ (𝐾‘{𝐶}) ⊆ 𝑈) ↔ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈))
37	2, 17, 36	mpbi2and 958	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈)
38	27	lsmub1 17894	. . . . . 6 ⊢ (((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺)) → (𝑆 ⊕ 𝑊) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
39	30, 34, 38	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
40	27	lsmub2 17895	. . . . . . 7 ⊢ (((𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺)) → (𝐾‘{𝐶}) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
41	30, 34, 40	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐾‘{𝐶}) ⊆ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
42	32	snssd 4281	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ 𝐵)
43	9, 15, 42	mrcssidd 16108	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ (𝐾‘{𝐶}))
44		snssg 4268	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐵 → (𝐶 ∈ (𝐾‘{𝐶}) ↔ {𝐶} ⊆ (𝐾‘{𝐶})))
45	32, 44	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝐶 ∈ (𝐾‘{𝐶}) ↔ {𝐶} ⊆ (𝐾‘{𝐶})))
46	43, 45	mpbird 246	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ (𝐾‘{𝐶}))
47	41, 46	sseldd 3569	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
48		eldifn 3695	. . . . . 6 ⊢ (𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)) → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
49	48	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ¬ 𝐶 ∈ (𝑆 ⊕ 𝑊))
50	39, 47, 49	ssnelpssd 3681	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
51	27	lsmub1 17894	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑊))
52	25, 26, 51	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑊))
53	22	snssd 4281	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
54	8, 15, 53	mrcssidd 16108	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
55	54, 18	syl6sseqr 3615	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
56		snssg 4268	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑈 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
57	21, 56	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ↔ {𝐴} ⊆ 𝑆))
58	55, 57	mpbird 246	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
59	52, 58	sseldd 3569	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝑆 ⊕ 𝑊))
60	59	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐴 ∈ (𝑆 ⊕ 𝑊))
61	39, 60	sseldd 3569	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
62	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐺 ∈ Abel)
63	27	lsmsubg2 18085	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ (𝑆 ⊕ 𝑊) ∈ (SubGrp‘𝐺) ∧ (𝐾‘{𝐶}) ∈ (SubGrp‘𝐺)) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ∈ (SubGrp‘𝐺))
64	62, 30, 34, 63	syl3anc 1318	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ∈ (SubGrp‘𝐺))
65		pgpfac1.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))
66	65	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤))
67		psseq1 3656	. . . . . . . . 9 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (𝑤 ⊊ 𝑈 ↔ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈))
68		eleq2 2677	. . . . . . . . 9 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
69	67, 68	anbi12d 743	. . . . . . . 8 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → ((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) ↔ (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ∧ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))))
70		psseq2 3657	. . . . . . . . 9 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → ((𝑆 ⊕ 𝑊) ⊊ 𝑤 ↔ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
71	70	notbid 307	. . . . . . . 8 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤 ↔ ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
72	69, 71	imbi12d 333	. . . . . . 7 ⊢ (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})))))
73	72	rspcv 3278	. . . . . 6 ⊢ (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ∈ (SubGrp‘𝐺) → (∀𝑤 ∈ (SubGrp‘𝐺)((𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤) → ¬ (𝑆 ⊕ 𝑊) ⊊ 𝑤) → ((((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ∧ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))) → ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))))
74	64, 66, 73	sylc 63	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ∧ 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))) → ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
75	61, 74	mpan2d 706	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 → ¬ (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶}))))
76	50, 75	mt2d 130	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ¬ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈)
77		npss 3679	. . 3 ⊢ (¬ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈 ↔ (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈 → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈))
78	76, 77	sylib 207	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊆ 𝑈 → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈))
79	37, 78	mpd 15	1 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) = 𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540   ⊊ wpss 3541  {csn 4125   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  Basecbs 15695  0_gc0g 15923  Moorecmre 16065  mrClscmrc 16066  ACScacs 16068  Grpcgrp 17245  SubGrpcsubg 17411  odcod 17767  gExcgex 17768   pGrp cpgp 17769  LSSumclsm 17872  Abelcabl 18017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-cntz 17573  df-lsm 17874  df-cmn 18018  df-abl 18019

This theorem is referenced by:  pgpfac1lem2  18297  pgpfac1lem3  18299