ghmqusnsglem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ghmqusnsglem2		Structured version Visualization version GIF version

Theorem ghmqusnsglem2 19251

Description: Lemma for ghmqusnsg 19252. (Contributed by Thierry Arnoux, 13-May-2025.)

**Hypotheses**
Ref	Expression
ghmqusnsg.0	⊢ 0 = (0_g‘𝐻)
ghmqusnsg.f	⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
ghmqusnsg.k	⊢ 𝐾 = (^◡𝐹 “ { 0 })
ghmqusnsg.q	⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝑁))
ghmqusnsg.j	⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
ghmqusnsg.n	⊢ (𝜑 → 𝑁 ⊆ 𝐾)
ghmqusnsg.1	⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
ghmqusnsglem2.y	⊢ (𝜑 → 𝑌 ∈ (Base‘𝑄))

**Assertion**
Ref	Expression
ghmqusnsglem2	⊢ (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥))

Distinct variable groups: 𝐹,𝑞 𝐺,𝑞,𝑥 𝐾,𝑞,𝑥 𝑁,𝑞 𝑄,𝑞 𝜑,𝑞,𝑥 𝑥,𝑁 𝑌,𝑞,𝑥 Allowed substitution hints: 𝑄(𝑥) 𝐹(𝑥) 𝐻(𝑥,𝑞) 𝐽(𝑥,𝑞) 0 (𝑥,𝑞)

**Proof of Theorem ghmqusnsglem2**
Step	Hyp	Ref	Expression
1		ghmqusnsglem2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑄))
2		ghmqusnsg.q	. . . . . 6 ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝑁))
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝑁)))
4		eqidd 2742	. . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5		ovexd 7395	. . . . 5 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) ∈ V)
6		ghmqusnsg.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
7		ghmgrp1 19188	. . . . . 6 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
9	3, 4, 5, 8	qusbas 17504	. . . 4 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) = (Base‘𝑄))
10	1, 9	eleqtrrd 2844	. . 3 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁)))
11		elqsg 8704	. . . 4 ⊢ (𝑌 ∈ (Base‘𝑄) → (𝑌 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) ↔ ∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~_QG 𝑁)))
12	11	biimpa 478	. . 3 ⊢ ((𝑌 ∈ (Base‘𝑄) ∧ 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁))) → ∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~_QG 𝑁))
13	1, 10, 12	syl2anc 591	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~_QG 𝑁))
14		ghmqusnsg.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
15		nsgsubg 19128	. . . . . . . . 9 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
16		eqid 2741	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2741	. . . . . . . . . 10 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
18	16, 17	eqger 19148	. . . . . . . . 9 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~_QG 𝑁) Er (Base‘𝐺))
19	14, 15, 18	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) Er (Base‘𝐺))
20	19	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝐺 ~_QG 𝑁) Er (Base‘𝐺))
21		simplr 775	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑥 ∈ (Base‘𝐺))
22		ecref 8683	. . . . . . 7 ⊢ (((𝐺 ~_QG 𝑁) Er (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ [𝑥](𝐺 ~_QG 𝑁))
23	20, 21, 22	syl2anc 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑥 ∈ [𝑥](𝐺 ~_QG 𝑁))
24		simpr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑌 = [𝑥](𝐺 ~_QG 𝑁))
25	23, 24	eleqtrrd 2844	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑥 ∈ 𝑌)
26	24	fveq2d 6835	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝐽‘𝑌) = (𝐽‘[𝑥](𝐺 ~_QG 𝑁)))
27		ghmqusnsg.0	. . . . . . 7 ⊢ 0 = (0_g‘𝐻)
28	6	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
29		ghmqusnsg.k	. . . . . . 7 ⊢ 𝐾 = (^◡𝐹 “ { 0 })
30		ghmqusnsg.j	. . . . . . 7 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))
31		ghmqusnsg.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ⊆ 𝐾)
32	31	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑁 ⊆ 𝐾)
33	14	ad2antrr 733	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
34	27, 28, 29, 2, 30, 32, 33, 21	ghmqusnsglem1 19250	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝐽‘[𝑥](𝐺 ~_QG 𝑁)) = (𝐹‘𝑥))
35	26, 34	eqtrd 2776	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝐽‘𝑌) = (𝐹‘𝑥))
36	25, 35	jca 517	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝑥 ∈ 𝑌 ∧ (𝐽‘𝑌) = (𝐹‘𝑥)))
37	36	expl 459	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐺) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝑥 ∈ 𝑌 ∧ (𝐽‘𝑌) = (𝐹‘𝑥))))
38	37	reximdv2 3151	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~_QG 𝑁) → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥)))
39	13, 38	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ∃wrex 3065 Vcvv 3433 ⊆ wss 3885 {csn 4558 ∪ cuni 4841 ↦ cmpt 5156 ^◡ccnv 5620 “ cima 5624 ‘cfv 6489 (class class class)co 7360 Er wer 8634 [cec 8635 / cqs 8636 Basecbs 17174 0_gc0g 17397 /_s cqus 17464 Grpcgrp 18904 SubGrpcsubg 19091 NrmSGrpcnsg 19092 ~_QG cqg 19093 GrpHom cghm 19182

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-ec 8639 df-qs 8643 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-0g 17399 df-imas 17467 df-qus 17468 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-subg 19094 df-nsg 19095 df-eqg 19096 df-ghm 19183

This theorem is referenced by: ghmqusnsg 19252 rhmqusnsg 21282

W3C validator