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Mirrors > Home > ILE Home > Th. List > grpinvnz | 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 5517 | . . . . . 6 ⊢ ((𝑁‘𝑋) = 0 → (𝑁‘(𝑁‘𝑋)) = (𝑁‘ 0 )) | |
2 | 1 | adantl 277 | . . . . 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 12943 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
6 | 5 | adantr 276 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
7 | grpinvnzcl.z | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
8 | 7, 4 | grpinvid 12936 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑁‘ 0 ) = 0 ) |
9 | 8 | ad2antrr 488 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → (𝑁‘ 0 ) = 0 ) |
10 | 2, 6, 9 | 3eqtr3d 2218 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑁‘𝑋) = 0 ) → 𝑋 = 0 ) |
11 | 10 | ex 115 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) = 0 → 𝑋 = 0 )) |
12 | 11 | necon3d 2391 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 ≠ 0 → (𝑁‘𝑋) ≠ 0 )) |
13 | 12 | 3impia 1200 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑁‘𝑋) ≠ 0 ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ‘cfv 5218 Basecbs 12465 0gc0g 12711 Grpcgrp 12883 invgcminusg 12884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7905 ax-resscn 7906 ax-1re 7908 ax-addrcl 7911 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-inn 8923 df-2 8981 df-ndx 12468 df-slot 12469 df-base 12471 df-plusg 12552 df-0g 12713 df-mgm 12781 df-sgrp 12814 df-mnd 12824 df-grp 12886 df-minusg 12887 |
This theorem is referenced by: grpinvnzcl 12948 |
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