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Mirrors > Home > MPE Home > Th. List > ghmquskerlem2 | Structured version Visualization version GIF version |
Description: Lemma for ghmqusker 19229. (Contributed by Thierry Arnoux, 14-Feb-2025.) |
Ref | Expression |
---|---|
ghmqusker.1 | ⊢ 0 = (0g‘𝐻) |
ghmqusker.f | ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
ghmqusker.k | ⊢ 𝐾 = (◡𝐹 “ { 0 }) |
ghmqusker.q | ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) |
ghmqusker.j | ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) |
ghmquskerlem2.y | ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑄)) |
Ref | Expression |
---|---|
ghmquskerlem2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmquskerlem2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑄)) | |
2 | ghmqusker.q | . . . . . 6 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))) |
4 | eqidd 2728 | . . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | ovexd 7449 | . . . . 5 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V) | |
6 | ghmqusker.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | |
7 | ghmgrp1 19163 | . . . . . 6 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp) |
9 | 3, 4, 5, 8 | qusbas 17518 | . . . 4 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄)) |
10 | 1, 9 | eleqtrrd 2831 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾))) |
11 | elqsg 8778 | . . . 4 ⊢ (𝑌 ∈ (Base‘𝑄) → (𝑌 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ↔ ∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~QG 𝐾))) | |
12 | 11 | biimpa 476 | . . 3 ⊢ ((𝑌 ∈ (Base‘𝑄) ∧ 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾))) → ∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~QG 𝐾)) |
13 | 1, 10, 12 | syl2anc 583 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~QG 𝐾)) |
14 | ghmqusker.k | . . . . . . . . . 10 ⊢ 𝐾 = (◡𝐹 “ { 0 }) | |
15 | ghmqusker.1 | . . . . . . . . . . . 12 ⊢ 0 = (0g‘𝐻) | |
16 | 15 | ghmker 19187 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺)) |
17 | nsgsubg 19104 | . . . . . . . . . . 11 ⊢ ((◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺) → (◡𝐹 “ { 0 }) ∈ (SubGrp‘𝐺)) | |
18 | 6, 16, 17 | 3syl 18 | . . . . . . . . . 10 ⊢ (𝜑 → (◡𝐹 “ { 0 }) ∈ (SubGrp‘𝐺)) |
19 | 14, 18 | eqeltrid 2832 | . . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) |
20 | eqid 2727 | . . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
21 | eqid 2727 | . . . . . . . . . 10 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾) | |
22 | 20, 21 | eqger 19124 | . . . . . . . . 9 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺)) |
23 | 19, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺)) |
24 | 23 | ad2antrr 725 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → (𝐺 ~QG 𝐾) Er (Base‘𝐺)) |
25 | simplr 768 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → 𝑥 ∈ (Base‘𝐺)) | |
26 | ecref 8762 | . . . . . . 7 ⊢ (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ [𝑥](𝐺 ~QG 𝐾)) | |
27 | 24, 25, 26 | syl2anc 583 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → 𝑥 ∈ [𝑥](𝐺 ~QG 𝐾)) |
28 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → 𝑌 = [𝑥](𝐺 ~QG 𝐾)) | |
29 | 27, 28 | eleqtrrd 2831 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → 𝑥 ∈ 𝑌) |
30 | 28 | fveq2d 6895 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑌) = (𝐽‘[𝑥](𝐺 ~QG 𝐾))) |
31 | 6 | ad2antrr 725 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
32 | ghmqusker.j | . . . . . . 7 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) | |
33 | 15, 31, 14, 2, 32, 25 | ghmquskerlem1 19225 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹‘𝑥)) |
34 | 30, 33 | eqtrd 2767 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑌) = (𝐹‘𝑥)) |
35 | 29, 34 | jca 511 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → (𝑥 ∈ 𝑌 ∧ (𝐽‘𝑌) = (𝐹‘𝑥))) |
36 | 35 | expl 457 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐺) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → (𝑥 ∈ 𝑌 ∧ (𝐽‘𝑌) = (𝐹‘𝑥)))) |
37 | 36 | reximdv2 3159 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ (Base‘𝐺)𝑌 = [𝑥](𝐺 ~QG 𝐾) → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥))) |
38 | 13, 37 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 Vcvv 3469 {csn 4624 ∪ cuni 4903 ↦ cmpt 5225 ◡ccnv 5671 “ cima 5675 ‘cfv 6542 (class class class)co 7414 Er wer 8715 [cec 8716 / cqs 8717 Basecbs 17171 0gc0g 17412 /s cqus 17478 Grpcgrp 18881 SubGrpcsubg 19066 NrmSGrpcnsg 19067 ~QG cqg 19068 GrpHom cghm 19158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-ec 8720 df-qs 8724 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-0g 17414 df-imas 17481 df-qus 17482 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-subg 19069 df-nsg 19070 df-eqg 19071 df-ghm 19159 |
This theorem is referenced by: ghmquskerlem3 19228 ghmqusker 19229 lmhmqusker 33067 rhmquskerlem 33076 |
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