![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 5985 | . . . . . 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 3983 | . . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) |
7 | qusima.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺) | |
8 | 7 | subgss 19089 | . . . . . . . . 9 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝐵) |
9 | 6, 8 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ⊆ 𝐵) |
10 | 9 | resmptd 6049 | . . . . . . 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 19120 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺)) | |
14 | 12, 13 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (SubGrp‘𝐺)) |
15 | 14 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝑁 ∈ (SubGrp‘𝐺)) |
16 | 9 | sselda 3982 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → 𝑥 ∈ 𝐵) |
17 | 7, 11, 15, 16 | quslsm 33140 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐻) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} ⊕ 𝑁)) |
18 | 17 | mpteq2dva 5252 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐻 ↦ [𝑥](𝐺 ~QG 𝑁)) = (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
19 | 10, 18 | eqtrd 2768 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ↾ 𝐻) = (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
20 | 3, 19 | eqtr2id 2781 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁)) = (𝐹 ↾ 𝐻)) |
21 | 20 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐻) → (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁)) = (𝐹 ↾ 𝐻)) |
22 | 21 | rneqd 5944 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ran (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁)) = ran (𝐹 ↾ 𝐻)) |
23 | mpteq1 5245 | . . . . 5 ⊢ (ℎ = 𝐻 → (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) | |
24 | 23 | rneqd 5944 | . . . 4 ⊢ (ℎ = 𝐻 → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
25 | 24 | adantl 480 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = ran (𝑥 ∈ 𝐻 ↦ ({𝑥} ⊕ 𝑁))) |
26 | df-ima 5695 | . . . 4 ⊢ (𝐹 “ 𝐻) = ran (𝐹 ↾ 𝐻) | |
27 | 26 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ ℎ = 𝐻) → (𝐹 “ 𝐻) = ran (𝐹 ↾ 𝐻)) |
28 | 22, 25, 27 | 3eqtr4d 2778 | . 2 ⊢ ((𝜑 ∧ ℎ = 𝐻) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) = (𝐹 “ 𝐻)) |
29 | 7 | fvexi 6916 | . . . . 5 ⊢ 𝐵 ∈ V |
30 | 29 | mptex 7241 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁)) ∈ V |
31 | 2, 30 | eqeltri 2825 | . . 3 ⊢ 𝐹 ∈ V |
32 | imaexg 7927 | . . 3 ⊢ (𝐹 ∈ V → (𝐹 “ 𝐻) ∈ V) | |
33 | 31, 32 | mp1i 13 | . 2 ⊢ (𝜑 → (𝐹 “ 𝐻) ∈ V) |
34 | 1, 28, 5, 33 | fvmptd2 7018 | 1 ⊢ (𝜑 → (𝐸‘𝐻) = (𝐹 “ 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 {csn 4632 ↦ cmpt 5235 ran crn 5683 ↾ cres 5684 “ cima 5685 ‘cfv 6553 (class class class)co 7426 [cec 8729 Basecbs 17187 /s cqus 17494 SubGrpcsubg 19082 NrmSGrpcnsg 19083 ~QG cqg 19084 LSSumclsm 19596 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-ec 8733 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-subg 19085 df-nsg 19086 df-eqg 19087 df-oppg 19304 df-lsm 19598 |
This theorem is referenced by: qusrn 33144 nsgmgc 33147 |
Copyright terms: Public domain | W3C validator |