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Theorem ghmqusnsglem2 19269

Description: Lemma for ghmqusnsg 19270. (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 2737	. . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5		ovexd 7445	. . . . 5 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) ∈ V)
6		ghmqusnsg.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
7		ghmgrp1 19206	. . . . . 6 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
9	3, 4, 5, 8	qusbas 17564	. . . 4 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) = (Base‘𝑄))
10	1, 9	eleqtrrd 2838	. . 3 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁)))
11		elqsg 8787	. . . 4 ⊢ (𝑌 ∈ (Base‘𝑄) → (𝑌 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) ↔ ∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~_QG 𝑁)))
12	11	biimpa 476	. . 3 ⊢ ((𝑌 ∈ (Base‘𝑄) ∧ 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁))) → ∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~_QG 𝑁))
13	1, 10, 12	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~_QG 𝑁))
14		ghmqusnsg.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))
15		nsgsubg 19146	. . . . . . . . 9 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
16		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2736	. . . . . . . . . 10 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
18	16, 17	eqger 19166	. . . . . . . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝐺 ~_QG 𝑁) Er (Base‘𝐺))
21		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑥 ∈ (Base‘𝐺))
22		ecref 8769	. . . . . . 7 ⊢ (((𝐺 ~_QG 𝑁) Er (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ [𝑥](𝐺 ~_QG 𝑁))
23	20, 21, 22	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑥 ∈ [𝑥](𝐺 ~_QG 𝑁))
24		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑌 = [𝑥](𝐺 ~_QG 𝑁))
25	23, 24	eleqtrrd 2838	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑥 ∈ 𝑌)
26	24	fveq2d 6885	. . . . . 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 726	. . . . . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑁 ⊆ 𝐾)
33	14	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
34	27, 28, 29, 2, 30, 32, 33, 21	ghmqusnsglem1 19268	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝐽‘[𝑥](𝐺 ~_QG 𝑁)) = (𝐹‘𝑥))
35	26, 34	eqtrd 2771	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝐽‘𝑌) = (𝐹‘𝑥))
36	25, 35	jca 511	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝑥 ∈ 𝑌 ∧ (𝐽‘𝑌) = (𝐹‘𝑥)))
37	36	expl 457	. . 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 395 = wceq 1540 ∈ wcel 2109 ∃wrex 3061 Vcvv 3464 ⊆ wss 3931 {csn 4606 ∪ cuni 4888 ↦ cmpt 5206 ^◡ccnv 5658 “ cima 5662 ‘cfv 6536 (class class class)co 7410 Er wer 8721 [cec 8722 / cqs 8723 Basecbs 17233 0_gc0g 17458 /_s cqus 17524 Grpcgrp 18921 SubGrpcsubg 19108 NrmSGrpcnsg 19109 ~_QG cqg 19110 GrpHom cghm 19200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-ec 8726 df-qs 8730 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-0g 17460 df-imas 17527 df-qus 17528 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-grp 18924 df-minusg 18925 df-subg 19111 df-nsg 19112 df-eqg 19113 df-ghm 19201

This theorem is referenced by: ghmqusnsg 19270 rhmqusnsg 21251

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