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Mirrors > Home > MPE Home > Th. List > grpinv11 | Structured version Visualization version GIF version |
Description: The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.) |
Ref | Expression |
---|---|
grpinvinv.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvinv.n | ⊢ 𝑁 = (invg‘𝐺) |
grpinv11.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
grpinv11.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
grpinv11.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
grpinv11 | ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6446 | . . . . 5 ⊢ ((𝑁‘𝑋) = (𝑁‘𝑌) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘(𝑁‘𝑌))) | |
2 | 1 | adantl 475 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘(𝑁‘𝑌))) |
3 | grpinv11.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
4 | grpinv11.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | grpinvinv.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
6 | grpinvinv.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝐺) | |
7 | 5, 6 | grpinvinv 17869 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
8 | 3, 4, 7 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
9 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
10 | grpinv11.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | 5, 6 | grpinvinv 17869 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
12 | 3, 10, 11 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
13 | 12 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
14 | 2, 9, 13 | 3eqtr3d 2821 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → 𝑋 = 𝑌) |
15 | 14 | ex 403 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) → 𝑋 = 𝑌)) |
16 | fveq2 6446 | . 2 ⊢ (𝑋 = 𝑌 → (𝑁‘𝑋) = (𝑁‘𝑌)) | |
17 | 15, 16 | impbid1 217 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 Basecbs 16255 Grpcgrp 17809 invgcminusg 17810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 |
This theorem is referenced by: gexdvds 18383 dchrisum0re 25654 mapdpglem30 37840 |
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