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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qusima | Structured version Visualization version GIF version |
Description: The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024.) |
Ref | Expression |
---|---|
qusima.b | ⊢ 𝐵 = (Base‘𝐺) |
qusima.q | ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) |
qusima.p | ⊢ ⊕ = (LSSum‘𝐺) |
qusima.e | ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
qusima.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) |
qusima.n | ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) |
qusima.h | ⊢ (𝜑 → 𝐻 ∈ 𝑆) |
qusima.s | ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝐺)) |
Ref | Expression |
---|---|
qusima | ⊢ (𝜑 → (𝐸‘𝐻) = (𝐹 “ 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusima.e | . 2 ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) | |
2 | qusima.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) | |
3 | 2 | reseq1i 5934 | . . . . . 6 ⊢ (𝐹 ↾ 𝐻) = ((𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ↾ 𝐻) |
4 | qusima.s | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝐺)) | |
5 | qusima.h | . . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ 𝑆) | |
6 | 4, 5 | sseldd 3946 | . . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) |
7 | qusima.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺) | |
8 | 7 | subgss 18934 | . . . . . . . . 9 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝐵) |
9 | 6, 8 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ 𝐵) |
10 | 9 | resmptd 5995 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ↾ 𝐻) = (𝑥 ∈ 𝐻 ↦ [𝑥](𝐺 ~QG 𝑁))) |
11 | qusima.p | . . . . . . . . 9 ⊢ ⊕ = (LSSum‘𝐺) | |
12 | qusima.n | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) | |
13 | nsgsubg 18965 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺)) | |
14 | 12, 13 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺)) |
15 | 14 | adantr 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝑁 ∈ (SubGrp‘𝐺)) |
16 | 9 | sselda 3945 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝑥 ∈ 𝐵) |
17 | 7, 11, 15, 16 | quslsm 32234 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁)) |
18 | 17 | mpteq2dva 5206 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐻 ↦ [𝑥](𝐺 ~QG 𝑁)) = (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
19 | 10, 18 | eqtrd 2773 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ↾ 𝐻) = (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
20 | 3, 19 | eqtr2id 2786 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁)) = (𝐹 ↾ 𝐻)) |
21 | 20 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐻) → (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁)) = (𝐹 ↾ 𝐻)) |
22 | 21 | rneqd 5894 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ran (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁)) = ran (𝐹 ↾ 𝐻)) |
23 | mpteq1 5199 | . . . . 5 ⊢ (ℎ = 𝐻 → (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) | |
24 | 23 | rneqd 5894 | . . . 4 ⊢ (ℎ = 𝐻 → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
25 | 24 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
26 | df-ima 5647 | . . . 4 ⊢ (𝐹 “ 𝐻) = ran (𝐹 ↾ 𝐻) | |
27 | 26 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐻) → (𝐹 “ 𝐻) = ran (𝐹 ↾ 𝐻)) |
28 | 22, 25, 27 | 3eqtr4d 2783 | . 2 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝐹 “ 𝐻)) |
29 | 7 | fvexi 6857 | . . . . 5 ⊢ 𝐵 ∈ V |
30 | 29 | mptex 7174 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ∈ V |
31 | 2, 30 | eqeltri 2830 | . . 3 ⊢ 𝐹 ∈ V |
32 | imaexg 7853 | . . 3 ⊢ (𝐹 ∈ V → (𝐹 “ 𝐻) ∈ V) | |
33 | 31, 32 | mp1i 13 | . 2 ⊢ (𝜑 → (𝐹 “ 𝐻) ∈ V) |
34 | 1, 28, 5, 33 | fvmptd2 6957 | 1 ⊢ (𝜑 → (𝐸‘𝐻) = (𝐹 “ 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 {csn 4587 ↦ cmpt 5189 ran crn 5635 ↾ cres 5636 “ cima 5637 ‘cfv 6497 (class class class)co 7358 [cec 8649 Basecbs 17088 /s cqus 17392 SubGrpcsubg 18927 NrmSGrpcnsg 18928 ~QG cqg 18929 LSSumclsm 19421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-ec 8653 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-subg 18930 df-nsg 18931 df-eqg 18932 df-oppg 19129 df-lsm 19423 |
This theorem is referenced by: nsgmgc 32238 |
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