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| Description: Lemma for ghmqusnsg 19301. (Contributed by Thierry Arnoux, 13-May-2025.) | 
| Ref | Expression | 
|---|---|
| ghmqusnsg.0 | ⊢ 0 = (0g‘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)) | 
| Ref | Expression | 
|---|---|
| ghmqusnsglem2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥)) | 
| 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 7467 | . . . . 5 ⊢ (𝜑 → (𝐺 ~QG 𝑁) ∈ V) | |
| 6 | ghmqusnsg.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | |
| 7 | ghmgrp1 19237 | . . . . . 6 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp) | |
| 8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp) | 
| 9 | 3, 4, 5, 8 | qusbas 17591 | . . . 4 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) = (Base‘𝑄)) | 
| 10 | 1, 9 | eleqtrrd 2843 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁))) | 
| 11 | elqsg 8809 | . . . 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 19177 | . . . . . . . . 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 19197 | . . . . . . . . 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 8791 | . . . . . . 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 2843 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ 𝑌) | 
| 26 | 24 | fveq2d 6909 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝑁)) → (𝐽‘𝑌) = (𝐽‘[𝑥](𝐺 ~QG 𝑁))) | 
| 27 | ghmqusnsg.0 | . . . . . . 7 ⊢ 0 = (0g‘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 19299 | . . . . . 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 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 3163 | . 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 1539 ∈ wcel 2107 ∃wrex 3069 Vcvv 3479 ⊆ wss 3950 {csn 4625 ∪ cuni 4906 ↦ cmpt 5224 ◡ccnv 5683 “ cima 5687 ‘cfv 6560 (class class class)co 7432 Er wer 8743 [cec 8744 / cqs 8745 Basecbs 17248 0gc0g 17485 /s cqus 17551 Grpcgrp 18952 SubGrpcsubg 19139 NrmSGrpcnsg 19140 ~QG cqg 19141 GrpHom cghm 19231 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17487 df-imas 17554 df-qus 17555 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-subg 19142 df-nsg 19143 df-eqg 19144 df-ghm 19232 | 
| This theorem is referenced by: ghmqusnsg 19301 rhmqusnsg 21296 | 
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