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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > imasgim | Structured version Visualization version GIF version |
Description: A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.) |
Ref | Expression |
---|---|
imasgim.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasgim.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasgim.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) |
imasgim.r | ⊢ (𝜑 → 𝑅 ∈ Grp) |
Ref | Expression |
---|---|
imasgim | ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpIso 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2726 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
3 | eqid 2726 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2726 | . . 3 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
5 | imasgim.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
6 | imasgim.u | . . . . 5 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
7 | imasgim.v | . . . . 5 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
8 | eqidd 2727 | . . . . 5 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅)) | |
9 | imasgim.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
10 | f1ofo 6833 | . . . . . 6 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
12 | 9 | f1ocpbl 17477 | . . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑐) ∧ (𝐹‘𝑏) = (𝐹‘𝑑)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑐(+g‘𝑅)𝑑)))) |
13 | eqid 2726 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | 6, 7, 8, 11, 12, 5, 13 | imasgrp 18981 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑈))) |
15 | 14 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑈 ∈ Grp) |
16 | 6, 7, 11, 5 | imasbas 17464 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
17 | f1oeq3 6816 | . . . . . . 7 ⊢ (𝐵 = (Base‘𝑈) → (𝐹:𝑉–1-1-onto→𝐵 ↔ 𝐹:𝑉–1-1-onto→(Base‘𝑈))) | |
18 | 16, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐹:𝑉–1-1-onto→𝐵 ↔ 𝐹:𝑉–1-1-onto→(Base‘𝑈))) |
19 | 9, 18 | mpbid 231 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→(Base‘𝑈)) |
20 | 7 | f1oeq2d 6822 | . . . . 5 ⊢ (𝜑 → (𝐹:𝑉–1-1-onto→(Base‘𝑈) ↔ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈))) |
21 | 19, 20 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈)) |
22 | f1of 6826 | . . . 4 ⊢ (𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈) → 𝐹:(Base‘𝑅)⟶(Base‘𝑈)) | |
23 | 21, 22 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶(Base‘𝑈)) |
24 | 7 | eleq2d 2813 | . . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝑉 ↔ 𝑎 ∈ (Base‘𝑅))) |
25 | 7 | eleq2d 2813 | . . . . . 6 ⊢ (𝜑 → (𝑏 ∈ 𝑉 ↔ 𝑏 ∈ (Base‘𝑅))) |
26 | 24, 25 | anbi12d 630 | . . . . 5 ⊢ (𝜑 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)))) |
27 | 11, 12, 6, 7, 5, 3, 4 | imasaddval 17484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏))) |
28 | 27 | eqcomd 2732 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏))) |
29 | 28 | 3expib 1119 | . . . . 5 ⊢ (𝜑 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)))) |
30 | 26, 29 | sylbird 260 | . . . 4 ⊢ (𝜑 → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)))) |
31 | 30 | imp 406 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏))) |
32 | 1, 2, 3, 4, 5, 15, 23, 31 | isghmd 19147 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑈)) |
33 | 1, 2 | isgim 19184 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑈) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈))) |
34 | 32, 21, 33 | sylanbrc 582 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpIso 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⟶wf 6532 –onto→wfo 6534 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 +gcplusg 17203 0gc0g 17391 “s cimas 17456 Grpcgrp 18860 GrpHom cghm 19135 GrpIso cgim 19179 |
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-tp 4628 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-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 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-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17151 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-0g 17393 df-imas 17460 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-ghm 19136 df-gim 19181 |
This theorem is referenced by: isnumbasgrplem1 42403 |
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