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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 5970 | . . . . . 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 3978 | . . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) |
7 | qusima.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺) | |
8 | 7 | subgss 19052 | . . . . . . . . 9 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝐵) |
9 | 6, 8 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ 𝐵) |
10 | 9 | resmptd 6033 | . . . . . . 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 19083 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺)) | |
14 | 12, 13 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺)) |
15 | 14 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝑁 ∈ (SubGrp‘𝐺)) |
16 | 9 | sselda 3977 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝑥 ∈ 𝐵) |
17 | 7, 11, 15, 16 | quslsm 33022 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁)) |
18 | 17 | mpteq2dva 5241 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐻 ↦ [𝑥](𝐺 ~QG 𝑁)) = (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
19 | 10, 18 | eqtrd 2766 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ↾ 𝐻) = (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
20 | 3, 19 | eqtr2id 2779 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁)) = (𝐹 ↾ 𝐻)) |
21 | 20 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐻) → (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁)) = (𝐹 ↾ 𝐻)) |
22 | 21 | rneqd 5930 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ran (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁)) = ran (𝐹 ↾ 𝐻)) |
23 | mpteq1 5234 | . . . . 5 ⊢ (ℎ = 𝐻 → (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) | |
24 | 23 | rneqd 5930 | . . . 4 ⊢ (ℎ = 𝐻 → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
25 | 24 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
26 | df-ima 5682 | . . . 4 ⊢ (𝐹 “ 𝐻) = ran (𝐹 ↾ 𝐻) | |
27 | 26 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐻) → (𝐹 “ 𝐻) = ran (𝐹 ↾ 𝐻)) |
28 | 22, 25, 27 | 3eqtr4d 2776 | . 2 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝐹 “ 𝐻)) |
29 | 7 | fvexi 6898 | . . . . 5 ⊢ 𝐵 ∈ V |
30 | 29 | mptex 7219 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ∈ V |
31 | 2, 30 | eqeltri 2823 | . . 3 ⊢ 𝐹 ∈ V |
32 | imaexg 7902 | . . 3 ⊢ (𝐹 ∈ V → (𝐹 “ 𝐻) ∈ V) | |
33 | 31, 32 | mp1i 13 | . 2 ⊢ (𝜑 → (𝐹 “ 𝐻) ∈ V) |
34 | 1, 28, 5, 33 | fvmptd2 6999 | 1 ⊢ (𝜑 → (𝐸‘𝐻) = (𝐹 “ 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 {csn 4623 ↦ cmpt 5224 ran crn 5670 ↾ cres 5671 “ cima 5672 ‘cfv 6536 (class class class)co 7404 [cec 8700 Basecbs 17151 /s cqus 17458 SubGrpcsubg 19045 NrmSGrpcnsg 19046 ~QG cqg 19047 LSSumclsm 19552 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-ec 8704 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-subg 19048 df-nsg 19049 df-eqg 19050 df-oppg 19260 df-lsm 19554 |
This theorem is referenced by: qusrn 33026 nsgmgc 33029 |
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