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Mirrors > Home > MPE Home > Th. List > mulgnn0z | Structured version Visualization version GIF version |
Description: A group multiple of the identity, for nonnegative 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 |
---|---|
mulgnn0z | ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11887 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | id 22 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ) | |
3 | mulgnn0z.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | mulgnn0z.o | . . . . . 6 ⊢ 0 = (0g‘𝐺) | |
5 | 3, 4 | mndidcl 17918 | . . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
6 | eqid 2798 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | mulgnn0z.t | . . . . . 6 ⊢ · = (.g‘𝐺) | |
8 | eqid 2798 | . . . . . 6 ⊢ seq1((+g‘𝐺), (ℕ × { 0 })) = seq1((+g‘𝐺), (ℕ × { 0 })) | |
9 | 3, 6, 7, 8 | mulgnn 18224 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 0 ∈ 𝐵) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) |
10 | 2, 5, 9 | syl2anr 599 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) |
11 | 3, 6, 4 | mndlid 17923 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
12 | 5, 11 | mpdan 686 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) = 0 ) |
13 | 12 | adantr 484 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
14 | simpr 488 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
15 | nnuz 12269 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
16 | 14, 15 | eleqtrdi 2900 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ≥‘1)) |
17 | 5 | adantr 484 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 0 ∈ 𝐵) |
18 | elfznn 12931 | . . . . . 6 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ) | |
19 | fvconst2g 6941 | . . . . . 6 ⊢ (( 0 ∈ 𝐵 ∧ 𝑥 ∈ ℕ) → ((ℕ × { 0 })‘𝑥) = 0 ) | |
20 | 17, 18, 19 | syl2an 598 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → ((ℕ × { 0 })‘𝑥) = 0 ) |
21 | 13, 16, 20 | seqid3 13410 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁) = 0 ) |
22 | 10, 21 | eqtrd 2833 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (𝑁 · 0 ) = 0 ) |
23 | oveq1 7142 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 · 0 ) = (0 · 0 )) | |
24 | 3, 4, 7 | mulg0 18223 | . . . . 5 ⊢ ( 0 ∈ 𝐵 → (0 · 0 ) = 0 ) |
25 | 5, 24 | syl 17 | . . . 4 ⊢ (𝐺 ∈ Mnd → (0 · 0 ) = 0 ) |
26 | 23, 25 | sylan9eqr 2855 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 = 0) → (𝑁 · 0 ) = 0 ) |
27 | 22, 26 | jaodan 955 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝑁 · 0 ) = 0 ) |
28 | 1, 27 | sylan2b 596 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 {csn 4525 × cxp 5517 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 ℕcn 11625 ℕ0cn0 11885 ℤ≥cuz 12231 ...cfz 12885 seqcseq 13364 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Mndcmnd 17903 .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-mulg 18217 |
This theorem is referenced by: mulgz 18247 mulgnn0ass 18255 odmodnn0 18660 mulgmhm 18941 srg1expzeq1 19282 lply1binomsc 20936 tsmsxp 22760 frobrhm 30910 |
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