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Mirrors > Home > MPE Home > Th. List > Mathboxes > ghmquskerlem2 | Structured version Visualization version GIF version |
Description: Lemma for ghmqusker 33004. (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 2725 | . . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | ovexd 7437 | . . . . 5 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V) | |
6 | ghmqusker.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | |
7 | ghmgrp1 19135 | . . . . . 6 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp) |
9 | 3, 4, 5, 8 | qusbas 17492 | . . . 4 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄)) |
10 | 1, 9 | eleqtrrd 2828 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾))) |
11 | elqsg 8759 | . . . 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 19159 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (◡𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺)) |
17 | nsgsubg 19077 | . . . . . . . . . . 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 2829 | . . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) |
20 | eqid 2724 | . . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
21 | eqid 2724 | . . . . . . . . . 10 ⊢ (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾) | |
22 | 20, 21 | eqger 19097 | . . . . . . . . 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 723 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → (𝐺 ~QG 𝐾) Er (Base‘𝐺)) |
25 | simplr 766 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → 𝑥 ∈ (Base‘𝐺)) | |
26 | ecref 32405 | . . . . . . 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 2828 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → 𝑥 ∈ 𝑌) |
30 | 28 | fveq2d 6886 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘𝑌) = (𝐽‘[𝑥](𝐺 ~QG 𝐾))) |
31 | 6 | ad2antrr 723 | . . . . . . 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 33000 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝐾)) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹‘𝑥)) |
34 | 30, 33 | eqtrd 2764 | . . . . 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 3156 | . 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 1533 ∈ wcel 2098 ∃wrex 3062 Vcvv 3466 {csn 4621 ∪ cuni 4900 ↦ cmpt 5222 ◡ccnv 5666 “ cima 5670 ‘cfv 6534 (class class class)co 7402 Er wer 8697 [cec 8698 / cqs 8699 Basecbs 17145 0gc0g 17386 /s cqus 17452 Grpcgrp 18855 SubGrpcsubg 19039 NrmSGrpcnsg 19040 ~QG cqg 19041 GrpHom cghm 19130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-ec 8702 df-qs 8706 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-0g 17388 df-imas 17455 df-qus 17456 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-nsg 19043 df-eqg 19044 df-ghm 19131 |
This theorem is referenced by: ghmquskerlem3 33003 ghmqusker 33004 lmhmqusker 33006 rhmquskerlem 33015 |
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