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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 6885 | . . . . 5 ⊢ ((𝑁‘𝑋) = (𝑁‘𝑌) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘(𝑁‘𝑌))) | |
2 | 1 | adantl 481 | . . . 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 18935 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
8 | 3, 4, 7 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
10 | grpinv11.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | 5, 6 | grpinvinv 18935 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
12 | 3, 10, 11 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → (𝑁‘(𝑁‘𝑌)) = 𝑌) |
14 | 2, 9, 13 | 3eqtr3d 2774 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘𝑋) = (𝑁‘𝑌)) → 𝑋 = 𝑌) |
15 | 14 | ex 412 | . 2 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) → 𝑋 = 𝑌)) |
16 | fveq2 6885 | . 2 ⊢ (𝑋 = 𝑌 → (𝑁‘𝑋) = (𝑁‘𝑌)) | |
17 | 15, 16 | impbid1 224 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 Basecbs 17153 Grpcgrp 18863 invgcminusg 18864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-riota 7361 df-ov 7408 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 |
This theorem is referenced by: eqg0subg 19122 gexdvds 19504 dchrisum0re 27401 mapdpglem30 41086 |
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