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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 6670 | . . . . 5 ⊢ ((𝑁‘𝑋) = (𝑁‘𝑌) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘(𝑁‘𝑌))) | |
2 | 1 | adantl 484 | . . . 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 18166 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
8 | 3, 4, 7 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
9 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
10 | grpinv11.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | 5, 6 | grpinvinv 18166 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
12 | 3, 10, 11 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
14 | 2, 9, 13 | 3eqtr3d 2864 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → 𝑋 = 𝑌) |
15 | 14 | ex 415 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) → 𝑋 = 𝑌)) |
16 | fveq2 6670 | . 2 ⊢ (𝑋 = 𝑌 → (𝑁‘𝑋) = (𝑁‘𝑌)) | |
17 | 15, 16 | impbid1 227 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 Basecbs 16483 Grpcgrp 18103 invgcminusg 18104 |
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-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 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-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 |
This theorem is referenced by: gexdvds 18709 dchrisum0re 26089 mapdpglem30 38853 |
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