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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 2610 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2610 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
3 | eqid 2610 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2610 | . . 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 2611 | . . . . 5 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅)) | |
9 | imasgim.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
10 | f1ofo 6057 | . . . . . 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 16008 | . . . . 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 2610 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | 6, 7, 8, 11, 12, 5, 13 | imasgrp 17354 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑈))) |
15 | 14 | simpld 474 | . . 3 ⊢ (𝜑 → 𝑈 ∈ Grp) |
16 | 6, 7, 11, 5 | imasbas 15995 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
17 | f1oeq3 6042 | . . . . . . 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 221 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→(Base‘𝑈)) |
20 | f1oeq2 6041 | . . . . . 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 221 | . . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈)) |
23 | f1of 6050 | . . . 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 2673 | . . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝑉 ↔ 𝑎 ∈ (Base‘𝑅))) |
26 | 7 | eleq2d 2673 | . . . . . 6 ⊢ (𝜑 → (𝑏 ∈ 𝑉 ↔ 𝑏 ∈ (Base‘𝑅))) |
27 | 25, 26 | anbi12d 743 | . . . . 5 ⊢ (𝜑 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)))) |
28 | 11, 12, 6, 7, 5, 3, 4 | imasaddval 16015 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏))) |
29 | 28 | eqcomd 2616 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏))) |
30 | 29 | 3expib 1260 | . . . . 5 ⊢ (𝜑 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)))) |
31 | 27, 30 | sylbird 249 | . . . 4 ⊢ (𝜑 → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)))) |
32 | 31 | imp 444 | . . 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 17492 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑈)) |
34 | 1, 2 | isgim 17527 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑈) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈))) |
35 | 33, 22, 34 | sylanbrc 695 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpIso 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⟶wf 5800 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 “s cimas 15987 Grpcgrp 17245 GrpHom cghm 17480 GrpIso cgim 17522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-0g 15925 df-imas 15991 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-ghm 17481 df-gim 17524 |
This theorem is referenced by: isnumbasgrplem1 36690 |
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