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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 6673 | . . . . . 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 18169 | . . . . . 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 18163 | . . . . . 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 2867 | . . . 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 3040 | . 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 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 Basecbs 16486 0gc0g 16716 Grpcgrp 18106 invgcminusg 18107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-riota 7117 df-ov 7162 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 |
This theorem is referenced by: grpinvnzcl 18174 |
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