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Mirrors > Home > MPE Home > Th. List > ghmqusnsglem2 | Structured version Visualization version GIF version |
Description: Lemma for ghmqusnsg 19313. (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 2736 | . . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | ovexd 7466 | . . . . 5 ⊢ (𝜑 → (𝐺 ~QG 𝑁) ∈ V) | |
6 | ghmqusnsg.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | |
7 | ghmgrp1 19249 | . . . . . 6 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ Grp) |
9 | 3, 4, 5, 8 | qusbas 17592 | . . . 4 ⊢ (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) = (Base‘𝑄)) |
10 | 1, 9 | eleqtrrd 2842 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁))) |
11 | elqsg 8807 | . . . 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 19189 | . . . . . . . . 9 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺)) | |
16 | eqid 2735 | . . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
17 | eqid 2735 | . . . . . . . . . 10 ⊢ (𝐺 ~QG 𝑁) = (𝐺 ~QG 𝑁) | |
18 | 16, 17 | eqger 19209 | . . . . . . . . 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 769 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ (Base‘𝐺)) | |
22 | ecref 8789 | . . . . . . 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 2842 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ 𝑌) |
26 | 24 | fveq2d 6911 | . . . . . 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 19311 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑌 = [𝑥](𝐺 ~QG 𝑁)) → (𝐽‘[𝑥](𝐺 ~QG 𝑁)) = (𝐹‘𝑥)) |
35 | 26, 34 | eqtrd 2775 | . . . . 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 3162 | . 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 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 {csn 4631 ∪ cuni 4912 ↦ cmpt 5231 ◡ccnv 5688 “ cima 5692 ‘cfv 6563 (class class class)co 7431 Er wer 8741 [cec 8742 / cqs 8743 Basecbs 17245 0gc0g 17486 /s cqus 17552 Grpcgrp 18964 SubGrpcsubg 19151 NrmSGrpcnsg 19152 ~QG cqg 19153 GrpHom cghm 19243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 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 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 |
This theorem is referenced by: ghmqusnsg 19313 rhmqusnsg 21313 |
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