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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 2821 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2821 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
3 | eqid 2821 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2821 | . . 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 2822 | . . . . 5 ⊢ (𝜑 → (+g‘𝑅) = (+g‘𝑅)) | |
9 | imasgim.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
10 | f1ofo 6622 | . . . . . 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 16798 | . . . . 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 2821 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | 6, 7, 8, 11, 12, 5, 13 | imasgrp 18215 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑈))) |
15 | 14 | simpld 497 | . . 3 ⊢ (𝜑 → 𝑈 ∈ Grp) |
16 | 6, 7, 11, 5 | imasbas 16785 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
17 | f1oeq3 6606 | . . . . . . 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 234 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→(Base‘𝑈)) |
20 | 7 | f1oeq2d 6611 | . . . . 5 ⊢ (𝜑 → (𝐹:𝑉–1-1-onto→(Base‘𝑈) ↔ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈))) |
21 | 19, 20 | mpbid 234 | . . . 4 ⊢ (𝜑 → 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈)) |
22 | f1of 6615 | . . . 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 2898 | . . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝑉 ↔ 𝑎 ∈ (Base‘𝑅))) |
25 | 7 | eleq2d 2898 | . . . . . 6 ⊢ (𝜑 → (𝑏 ∈ 𝑉 ↔ 𝑏 ∈ (Base‘𝑅))) |
26 | 24, 25 | anbi12d 632 | . . . . 5 ⊢ (𝜑 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)))) |
27 | 11, 12, 6, 7, 5, 3, 4 | imasaddval 16805 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏))) |
28 | 27 | eqcomd 2827 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏))) |
29 | 28 | 3expib 1118 | . . . . 5 ⊢ (𝜑 → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)))) |
30 | 26, 29 | sylbird 262 | . . . 4 ⊢ (𝜑 → ((𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)))) |
31 | 30 | imp 409 | . . 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 18367 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑈)) |
33 | 1, 2 | isgim 18402 | . 2 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑈) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑈) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑈))) |
34 | 32, 21, 33 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpIso 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⟶wf 6351 –onto→wfo 6353 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 “s cimas 16777 Grpcgrp 18103 GrpHom cghm 18355 GrpIso cgim 18397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-0g 16715 df-imas 16781 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-ghm 18356 df-gim 18399 |
This theorem is referenced by: isnumbasgrplem1 39721 |
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