Theorem pgpfac1lem1 20177

Description: Lemma for pgpfac1 20183. (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 19886	. . . . . 6 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
5		pgpfac1.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
6	5	subgacs 19258	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘𝐵))
7		acsmre 17733	. . . . . 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 4088	. . . . . 6 ⊢ (𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊)) → 𝐶 ∈ 𝑈)
11	10	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ 𝑈)
12	11	snssd 4757	. . . 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 17701	. . . 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 19224	. . . . . . . . . 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 3941	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
23	15	mrcsncl 17693	. . . . . . . 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 2870	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
26		pgpfac1.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (SubGrp‘𝐺))
27		pgpfac1.l	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝐺)
28	27	lsmsubg2 19960	. . . . . 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 3940	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑈) → 𝐶 ∈ 𝐵)
32	10, 31	sylan2 605	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ 𝐵)
33	15	mrcsncl 17693	. . . . 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 19765	. . . 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 19758	. . . . . 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 19759	. . . . . . 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 4757	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ 𝐵)
43	9, 15, 42	mrcssidd 17706	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → {𝐶} ⊆ (𝐾‘{𝐶}))
44		snssg 4754	. . . . . . . 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 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐶 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
48		eldifn 4089	. . . . . 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 4073	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → (𝑆 ⊕ 𝑊) ⊊ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
51	27	lsmub1 19758	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑊 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (𝑆 ⊕ 𝑊))
52	25, 26, 51	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ (𝑆 ⊕ 𝑊))
53	22	snssd 4757	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ 𝐵)
54	8, 15, 53	mrcssidd 17706	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ (𝐾‘{𝐴}))
55	54, 18	sseqtrrdi 3981	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ 𝑆)
56		snssg 4754	. . . . . . . . . 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 3941	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝑆 ⊕ 𝑊))
60	59	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐴 ∈ (𝑆 ⊕ 𝑊))
61	39, 60	sseldd 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝑈 ∖ (𝑆 ⊕ 𝑊))) → 𝐴 ∈ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})))
62		psseq1 4047	. . . . . . . 8 ⊢ (𝑤 = ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) → (𝑤 ⊊ 𝑈 ↔ ((𝑆 ⊕ 𝑊) ⊕ (𝐾‘{𝐶})) ⊊ 𝑈))
63		eleq2 2855	. . . . . . . 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 4048	. . . . . . . 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 19960	. . . . . . 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 3585	. . . . 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 4071	. . 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 1570   ∈ wcel 2146  ∀wral 3082   ∖ cdif 3905   ∩ cin 3907   ⊆ wss 3908   ⊊ wpss 3909  {csn 4594   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Fincfn 8952  Basecbs 17294  0_gc0g 17517  Moorecmre 17659  mrClscmrc 17660  ACScacs 17662  Grpcgrp 19031  SubGrpcsubg 19217  odcod 19625  gExcgex 19626   pGrp cpgp 19627  LSSumclsm 19735  Abelcabl 19882

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-ress 17316  df-plusg 17348  df-0g 17519  df-mre 17663  df-mrc 17664  df-acs 17666  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-submnd 18873  df-grp 19034  df-minusg 19035  df-subg 19220  df-cntz 19418  df-lsm 19737  df-cmn 19883  df-abl 19884

This theorem is used by:  pgpfac1lem2  20178  pgpfac1lem3  20180