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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 2622 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2622 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
3 | eqid 2622 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2622 | . . 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 2623 | . . . . 5 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅)) | |
9 | imasgim.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
10 | f1ofo 6144 | . . . . . 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 16185 | . . . . 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 2622 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | 6, 7, 8, 11, 12, 5, 13 | imasgrp 17531 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑈))) |
15 | 14 | simpld 475 | . . 3 ⊢ (𝜑 → 𝑈 ∈ Grp) |
16 | 6, 7, 11, 5 | imasbas 16172 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
17 | f1oeq3 6129 | . . . . . . 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 222 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→(Base‘𝑈)) |
20 | f1oeq2 6128 | . . . . . 6 ⊢ (𝑉 = (Base‘𝑅) → (𝐹:𝑉–1-1-onto→(Base‘𝑈) ↔ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈))) | |
21 | 7, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐹:𝑉–1-1-onto→(Base‘𝑈) ↔ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈))) |
22 | 19, 21 | mpbid 222 | . . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈)) |
23 | f1of 6137 | . . . 4 ⊢ (𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈) → 𝐹:(Base‘𝑅)⟶(Base‘𝑈)) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶(Base‘𝑈)) |
25 | 7 | eleq2d 2687 | . . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝑉 ↔ 𝑎 ∈ (Base‘𝑅))) |
26 | 7 | eleq2d 2687 | . . . . . 6 ⊢ (𝜑 → (𝑏 ∈ 𝑉 ↔ 𝑏 ∈ (Base‘𝑅))) |
27 | 25, 26 | anbi12d 747 | . . . . 5 ⊢ (𝜑 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)))) |
28 | 11, 12, 6, 7, 5, 3, 4 | imasaddval 16192 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏))) |
29 | 28 | eqcomd 2628 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏))) |
30 | 29 | 3expib 1268 | . . . . 5 ⊢ (𝜑 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)))) |
31 | 27, 30 | sylbird 250 | . . . 4 ⊢ (𝜑 → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)))) |
32 | 31 | imp 445 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏))) |
33 | 1, 2, 3, 4, 5, 15, 24, 32 | isghmd 17669 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑈)) |
34 | 1, 2 | isgim 17704 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑈) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈))) |
35 | 33, 22, 34 | sylanbrc 698 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpIso 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⟶wf 5884 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 0gc0g 16100 “s cimas 16164 Grpcgrp 17422 GrpHom cghm 17657 GrpIso cgim 17699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-0g 16102 df-imas 16168 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-ghm 17658 df-gim 17701 |
This theorem is referenced by: isnumbasgrplem1 37671 |
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