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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 12530 | . 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 18763 | . . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) | 
| 6 | eqid 2736 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 7 | mulgnn0z.t | . . . . . 6 ⊢ · = (.g‘𝐺) | |
| 8 | eqid 2736 | . . . . . 6 ⊢ seq1((+g‘𝐺), (ℕ × { 0 })) = seq1((+g‘𝐺), (ℕ × { 0 })) | |
| 9 | 3, 6, 7, 8 | mulgnn 19094 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 0 ∈ 𝐵) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) | 
| 10 | 2, 5, 9 | syl2anr 597 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) | 
| 11 | 3, 6, 4 | mndlid 18768 | . . . . . . 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 480 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → ( 0 (+g‘𝐺) 0 ) = 0 ) | 
| 14 | simpr 484 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
| 15 | nnuz 12922 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
| 16 | 14, 15 | eleqtrdi 2850 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ≥‘1)) | 
| 17 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 0 ∈ 𝐵) | 
| 18 | elfznn 13594 | . . . . . 6 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ) | |
| 19 | fvconst2g 7223 | . . . . . 6 ⊢ (( 0 ∈ 𝐵 ∧ 𝑥 ∈ ℕ) → ((ℕ × { 0 })‘𝑥) = 0 ) | |
| 20 | 17, 18, 19 | syl2an 596 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → ((ℕ × { 0 })‘𝑥) = 0 ) | 
| 21 | 13, 16, 20 | seqid3 14088 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁) = 0 ) | 
| 22 | 10, 21 | eqtrd 2776 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (𝑁 · 0 ) = 0 ) | 
| 23 | oveq1 7439 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 · 0 ) = (0 · 0 )) | |
| 24 | 3, 4, 7 | mulg0 19093 | . . . . 5 ⊢ ( 0 ∈ 𝐵 → (0 · 0 ) = 0 ) | 
| 25 | 5, 24 | syl 17 | . . . 4 ⊢ (𝐺 ∈ Mnd → (0 · 0 ) = 0 ) | 
| 26 | 23, 25 | sylan9eqr 2798 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 = 0) → (𝑁 · 0 ) = 0 ) | 
| 27 | 22, 26 | jaodan 959 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝑁 · 0 ) = 0 ) | 
| 28 | 1, 27 | sylan2b 594 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 {csn 4625 × cxp 5682 ‘cfv 6560 (class class class)co 7432 0cc0 11156 1c1 11157 ℕcn 12267 ℕ0cn0 12528 ℤ≥cuz 12879 ...cfz 13548 seqcseq 14043 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Mndcmnd 18748 .gcmg 19086 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-seq 14044 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mulg 19087 | 
| This theorem is referenced by: mulgz 19121 mulgnn0ass 19129 odmodnn0 19559 mulgmhm 19846 srg1expzeq1 20223 frobrhm 21595 psdmplcl 22167 psdmvr 22174 lply1binomsc 22316 tsmsxp 24164 aks6d1c1p6 42116 aks6d1c2lem3 42128 | 
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