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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 6669 | . . . . . 6 ⊢ ((𝑁‘𝑋) = 0 → (𝑁‘(𝑁‘𝑋)) = (𝑁‘ 0 )) | |
2 | 1 | adantl 484 | . . . . 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 18165 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
6 | 5 | adantr 483 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
7 | grpinvnzcl.z | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
8 | 7, 4 | grpinvid 18159 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
9 | 8 | ad2antrr 724 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → (𝑁‘ 0 ) = 0 ) |
10 | 2, 6, 9 | 3eqtr3d 2864 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → 𝑋 = 0 ) |
11 | 10 | ex 415 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) = 0 → 𝑋 = 0 )) |
12 | 11 | necon3d 3037 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 ≠ 0 → (𝑁‘𝑋) ≠ 0 )) |
13 | 12 | 3impia 1113 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑁‘𝑋) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6354 Basecbs 16482 0gc0g 16712 Grpcgrp 18102 invgcminusg 18103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-riota 7113 df-ov 7158 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 |
This theorem is referenced by: grpinvnzcl 18170 |
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