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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 2728 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2728 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
3 | eqid 2728 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2728 | . . 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 2729 | . . . . 5 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅)) | |
9 | imasgim.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
10 | f1ofo 6851 | . . . . . 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 17514 | . . . . 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 2728 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | 6, 7, 8, 11, 12, 5, 13 | imasgrp 19019 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑈))) |
15 | 14 | simpld 493 | . . 3 ⊢ (𝜑 → 𝑈 ∈ Grp) |
16 | 6, 7, 11, 5 | imasbas 17501 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
17 | f1oeq3 6834 | . . . . . . 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 6840 | . . . . 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 6844 | . . . 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 2815 | . . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝑉 ↔ 𝑎 ∈ (Base‘𝑅))) |
25 | 7 | eleq2d 2815 | . . . . . 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 17521 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏))) |
28 | 27 | eqcomd 2734 | . . . . . 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 259 | . . . 4 ⊢ (𝜑 → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)))) |
31 | 30 | imp 405 | . . 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 19186 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑈)) |
33 | 1, 2 | isgim 19223 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑈) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈))) |
34 | 32, 21, 33 | sylanbrc 581 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpIso 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⟶wf 6549 –onto→wfo 6551 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 0gc0g 17428 “s cimas 17493 Grpcgrp 18897 GrpHom cghm 19174 GrpIso cgim 19218 |
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-tp 4637 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-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 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-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-0g 17430 df-imas 17497 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-ghm 19175 df-gim 19220 |
This theorem is referenced by: isnumbasgrplem1 42556 |
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