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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 18113 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | 1 | adantr 483 | . . 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 18257 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
7 | 2, 6 | sylan 582 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
8 | simpll 765 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → 𝐺 ∈ Grp) | |
9 | nn0z 12008 | . . . . 5 ⊢ (-𝑁 ∈ ℕ0 → -𝑁 ∈ ℤ) | |
10 | 9 | adantl 484 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → -𝑁 ∈ ℤ) |
11 | 3, 5 | grpidcl 18134 | . . . . 5 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
12 | 11 | ad2antrr 724 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → 0 ∈ 𝐵) |
13 | eqid 2824 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
14 | 3, 4, 13 | mulgneg 18249 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ -𝑁 ∈ ℤ ∧ 0 ∈ 𝐵) → (--𝑁 · 0 ) = ((invg‘𝐺)‘(-𝑁 · 0 ))) |
15 | 8, 10, 12, 14 | syl3anc 1367 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → (--𝑁 · 0 ) = ((invg‘𝐺)‘(-𝑁 · 0 ))) |
16 | zcn 11989 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
17 | 16 | ad2antlr 725 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ) |
18 | 17 | negnegd 10991 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → --𝑁 = 𝑁) |
19 | 18 | oveq1d 7174 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → (--𝑁 · 0 ) = (𝑁 · 0 )) |
20 | 3, 4, 5 | mulgnn0z 18257 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ -𝑁 ∈ ℕ0) → (-𝑁 · 0 ) = 0 ) |
21 | 2, 20 | sylan 582 | . . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → (-𝑁 · 0 ) = 0 ) |
22 | 21 | fveq2d 6677 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → ((invg‘𝐺)‘(-𝑁 · 0 )) = ((invg‘𝐺)‘ 0 )) |
23 | 5, 13 | grpinvid 18163 | . . . . 5 ⊢ (𝐺 ∈ Grp → ((invg‘𝐺)‘ 0 ) = 0 ) |
24 | 23 | ad2antrr 724 | . . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → ((invg‘𝐺)‘ 0 ) = 0 ) |
25 | 22, 24 | eqtrd 2859 | . . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → ((invg‘𝐺)‘(-𝑁 · 0 )) = 0 ) |
26 | 15, 19, 25 | 3eqtr3d 2867 | . 2 ⊢ (((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) ∧ -𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
27 | elznn0 11999 | . . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) | |
28 | 27 | simprbi 499 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)) |
29 | 28 | adantl 484 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)) |
30 | 7, 26, 29 | mpjaodan 955 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝑁 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 -cneg 10874 ℕ0cn0 11900 ℤcz 11984 Basecbs 16486 0gc0g 16716 Mndcmnd 17914 Grpcgrp 18106 invgcminusg 18107 .gcmg 18227 |
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 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 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-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-seq 13373 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-mulg 18228 |
This theorem is referenced by: mulgmodid 18269 odmod 18677 gexdvdsi 18711 |
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