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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 6663 | . . . . 5 ⊢ ((𝑁‘𝑋) = (𝑁‘𝑌) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘(𝑁‘𝑌))) | |
2 | 1 | adantl 482 | . . . 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 18104 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
8 | 3, 4, 7 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
9 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
10 | grpinv11.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | 5, 6 | grpinvinv 18104 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
12 | 3, 10, 11 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
14 | 2, 9, 13 | 3eqtr3d 2861 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → 𝑋 = 𝑌) |
15 | 14 | ex 413 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) → 𝑋 = 𝑌)) |
16 | fveq2 6663 | . 2 ⊢ (𝑋 = 𝑌 → (𝑁‘𝑋) = (𝑁‘𝑌)) | |
17 | 15, 16 | impbid1 226 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 Basecbs 16471 Grpcgrp 18041 invgcminusg 18042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-riota 7103 df-ov 7148 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 |
This theorem is referenced by: gexdvds 18638 dchrisum0re 26016 mapdpglem30 38718 |
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