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Theorem ghmqusnsglem2 19193

Description: Lemma for ghmqusnsg 19194. (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 2732	. . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
5		ovexd 7381	. . . . 5 ⊢ (𝜑 → (𝐺 ~_QG 𝑁) ∈ V)
6		ghmqusnsg.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))
7		ghmgrp1 19130	. . . . . 6 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp)
9	3, 4, 5, 8	qusbas 17449	. . . 4 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~_QG 𝑁)) = (Base‘𝑄))
10	1, 9	eleqtrrd 2834	. . 3 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~_QG 𝑁)))
11		elqsg 8688	. . . 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 19070	. . . . . . . . 9 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
16		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
17		eqid 2731	. . . . . . . . . 10 ⊢ (𝐺 ~_QG 𝑁) = (𝐺 ~_QG 𝑁)
18	16, 17	eqger 19090	. . . . . . . . 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 8667	. . . . . . 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 2834	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → 𝑥 ∈ 𝑌)
26	24	fveq2d 6826	. . . . . 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 19192	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~_QG 𝑁)) → (𝐽‘[𝑥](𝐺 ~_QG 𝑁)) = (𝐹‘𝑥))
35	26, 34	eqtrd 2766	. . . . 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 3142	. 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 1541 ∈ wcel 2111 ∃wrex 3056 Vcvv 3436 ⊆ wss 3897 {csn 4573 ∪ cuni 4856 ↦ cmpt 5170 ^◡ccnv 5613 “ cima 5617 ‘cfv 6481 (class class class)co 7346 Er wer 8619 [cec 8620 / cqs 8621 Basecbs 17120 0_gc0g 17343 /_s cqus 17409 Grpcgrp 18846 SubGrpcsubg 19033 NrmSGrpcnsg 19034 ~_QG cqg 19035 GrpHom cghm 19124

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-0g 17345 df-imas 17412 df-qus 17413 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-grp 18849 df-minusg 18850 df-subg 19036 df-nsg 19037 df-eqg 19038 df-ghm 19125

This theorem is referenced by: ghmqusnsg 19194 rhmqusnsg 21222

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