![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > grpinvnz | Structured version Visualization version GIF version |
Description: The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
Ref | Expression |
---|---|
grpinvnzcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvnzcl.z | ⊢ 0 = (0g‘𝐺) |
grpinvnzcl.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvnz | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑁‘𝑋) ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6885 | . . . . . 6 ⊢ ((𝑁‘𝑋) = 0 → (𝑁‘(𝑁‘𝑋)) = (𝑁‘ 0 )) | |
2 | 1 | adantl 481 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → (𝑁‘(𝑁‘𝑋)) = (𝑁‘ 0 )) |
3 | grpinvnzcl.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
4 | grpinvnzcl.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝐺) | |
5 | 3, 4 | grpinvinv 18935 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
6 | 5 | adantr 480 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
7 | grpinvnzcl.z | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
8 | 7, 4 | grpinvid 18929 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
9 | 8 | ad2antrr 723 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → (𝑁‘ 0 ) = 0 ) |
10 | 2, 6, 9 | 3eqtr3d 2774 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → 𝑋 = 0 ) |
11 | 10 | ex 412 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) = 0 → 𝑋 = 0 )) |
12 | 11 | necon3d 2955 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 ≠ 0 → (𝑁‘𝑋) ≠ 0 )) |
13 | 12 | 3impia 1114 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑁‘𝑋) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ‘cfv 6537 Basecbs 17153 0gc0g 17394 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: grpinvnzcl 18940 |
Copyright terms: Public domain | W3C validator |