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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 11971 | . 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 18035 | . . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
6 | eqid 2738 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | mulgnn0z.t | . . . . . 6 ⊢ · = (.g‘𝐺) | |
8 | eqid 2738 | . . . . . 6 ⊢ seq1((+g‘𝐺), (ℕ × { 0 })) = seq1((+g‘𝐺), (ℕ × { 0 })) | |
9 | 3, 6, 7, 8 | mulgnn 18343 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 0 ∈ 𝐵) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) |
10 | 2, 5, 9 | syl2anr 600 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) |
11 | 3, 6, 4 | mndlid 18040 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
12 | 5, 11 | mpdan 687 | . . . . . 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 12356 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
16 | 14, 15 | eleqtrdi 2843 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ≥‘1)) |
17 | 5 | adantr 484 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 0 ∈ 𝐵) |
18 | elfznn 13020 | . . . . . 6 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ) | |
19 | fvconst2g 6968 | . . . . . 6 ⊢ (( 0 ∈ 𝐵 ∧ 𝑥 ∈ ℕ) → ((ℕ × { 0 })‘𝑥) = 0 ) | |
20 | 17, 18, 19 | syl2an 599 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → ((ℕ × { 0 })‘𝑥) = 0 ) |
21 | 13, 16, 20 | seqid3 13499 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁) = 0 ) |
22 | 10, 21 | eqtrd 2773 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (𝑁 · 0 ) = 0 ) |
23 | oveq1 7171 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 · 0 ) = (0 · 0 )) | |
24 | 3, 4, 7 | mulg0 18342 | . . . . 5 ⊢ ( 0 ∈ 𝐵 → (0 · 0 ) = 0 ) |
25 | 5, 24 | syl 17 | . . . 4 ⊢ (𝐺 ∈ Mnd → (0 · 0 ) = 0 ) |
26 | 23, 25 | sylan9eqr 2795 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 = 0) → (𝑁 · 0 ) = 0 ) |
27 | 22, 26 | jaodan 957 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝑁 · 0 ) = 0 ) |
28 | 1, 27 | sylan2b 597 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2113 {csn 4513 × cxp 5517 ‘cfv 6333 (class class class)co 7164 0cc0 10608 1c1 10609 ℕcn 11709 ℕ0cn0 11969 ℤ≥cuz 12317 ...cfz 12974 seqcseq 13453 Basecbs 16579 +gcplusg 16661 0gc0g 16809 Mndcmnd 18020 .gcmg 18335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 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 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-seq 13454 df-0g 16811 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mulg 18336 |
This theorem is referenced by: mulgz 18366 mulgnn0ass 18374 odmodnn0 18779 mulgmhm 19060 srg1expzeq1 19401 lply1binomsc 21075 tsmsxp 22899 frobrhm 31054 |
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