![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mulgz | Structured version Visualization version GIF version |
Description: A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.) |
Ref | Expression |
---|---|
mulgnn0z.b | ⊢ 𝐵 = (Base‘𝐺) |
mulgnn0z.t | ⊢ · = (.g‘𝐺) |
mulgnn0z.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
mulgz | ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 · 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 18102 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | 1 | adantr 484 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → 𝐺 ∈ Mnd) |
3 | mulgnn0z.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
4 | mulgnn0z.t | . . . 4 ⊢ · = (.g‘𝐺) | |
5 | mulgnn0z.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
6 | 3, 4, 5 | mulgnn0z 18246 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
7 | 2, 6 | sylan 583 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
8 | simpll 766 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → 𝐺 ∈ Grp) | |
9 | nn0z 11993 | . . . . 5 ⊢ (-𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ) | |
10 | 9 | adantl 485 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → -𝑁 ∈ ℤ) |
11 | 3, 5 | grpidcl 18123 | . . . . 5 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
12 | 11 | ad2antrr 725 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → 0 ∈ 𝐵) |
13 | eqid 2798 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
14 | 3, 4, 13 | mulgneg 18238 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ -𝑁 ∈ ℤ ∧ 0 ∈ 𝐵) → (--𝑁 · 0 ) = ((invg‘𝐺)‘(-𝑁 · 0 ))) |
15 | 8, 10, 12, 14 | syl3anc 1368 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → (--𝑁 · 0 ) = ((invg‘𝐺)‘(-𝑁 · 0 ))) |
16 | zcn 11974 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
17 | 16 | ad2antlr 726 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ) |
18 | 17 | negnegd 10977 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → --𝑁 = 𝑁) |
19 | 18 | oveq1d 7150 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → (--𝑁 · 0 ) = (𝑁 · 0 )) |
20 | 3, 4, 5 | mulgnn0z 18246 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ -𝑁 ∈ ℕ0) → (-𝑁 · 0 ) = 0 ) |
21 | 2, 20 | sylan 583 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → (-𝑁 · 0 ) = 0 ) |
22 | 21 | fveq2d 6649 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → ((invg‘𝐺)‘(-𝑁 · 0 )) = ((invg‘𝐺)‘ 0 )) |
23 | 5, 13 | grpinvid 18152 | . . . . 5 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
24 | 23 | ad2antrr 725 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → ((invg‘𝐺)‘ 0 ) = 0 ) |
25 | 22, 24 | eqtrd 2833 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → ((invg‘𝐺)‘(-𝑁 · 0 )) = 0 ) |
26 | 15, 19, 25 | 3eqtr3d 2841 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
27 | elznn0 11984 | . . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) | |
28 | 27 | simprbi 500 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)) |
29 | 28 | adantl 485 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)) |
30 | 7, 26, 29 | mpjaodan 956 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 -cneg 10860 ℕ0cn0 11885 ℤcz 11969 Basecbs 16475 0gc0g 16705 Mndcmnd 17903 Grpcgrp 18095 invgcminusg 18096 .gcmg 18216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-mulg 18217 |
This theorem is referenced by: mulgmodid 18258 odmod 18666 gexdvdsi 18700 |
Copyright terms: Public domain | W3C validator |