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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 6229 | . . . . . 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 17529 | . . . . . 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 17523 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
9 | 8 | ad2antrr 762 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → (𝑁‘ 0 ) = 0 ) |
10 | 2, 6, 9 | 3eqtr3d 2693 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → 𝑋 = 0 ) |
11 | 10 | ex 449 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) = 0 → 𝑋 = 0 )) |
12 | 11 | necon3d 2844 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 ≠ 0 → (𝑁‘𝑋) ≠ 0 )) |
13 | 12 | 3impia 1280 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑁‘𝑋) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ‘cfv 5926 Basecbs 15904 0gc0g 16147 Grpcgrp 17469 invgcminusg 17470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 |
This theorem is referenced by: grpinvnzcl 17534 |
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