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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 12263 | . 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 18428 | . . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
6 | eqid 2733 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | mulgnn0z.t | . . . . . 6 ⊢ · = (.g‘𝐺) | |
8 | eqid 2733 | . . . . . 6 ⊢ seq1((+g‘𝐺), (ℕ × { 0 })) = seq1((+g‘𝐺), (ℕ × { 0 })) | |
9 | 3, 6, 7, 8 | mulgnn 18736 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 0 ∈ 𝐵) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) |
10 | 2, 5, 9 | syl2anr 596 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) |
11 | 3, 6, 4 | mndlid 18433 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
12 | 5, 11 | mpdan 683 | . . . . . 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 12649 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
16 | 14, 15 | eleqtrdi 2844 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ≥‘1)) |
17 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 0 ∈ 𝐵) |
18 | elfznn 13313 | . . . . . 6 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ) | |
19 | fvconst2g 7097 | . . . . . 6 ⊢ (( 0 ∈ 𝐵 ∧ 𝑥 ∈ ℕ) → ((ℕ × { 0 })‘𝑥) = 0 ) | |
20 | 17, 18, 19 | syl2an 595 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → ((ℕ × { 0 })‘𝑥) = 0 ) |
21 | 13, 16, 20 | seqid3 13795 | . . . 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 7302 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 · 0 ) = (0 · 0 )) | |
24 | 3, 4, 7 | mulg0 18735 | . . . . 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 954 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝑁 · 0 ) = 0 ) |
28 | 1, 27 | sylan2b 593 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 843 = wceq 1537 ∈ wcel 2101 {csn 4564 × cxp 5589 ‘cfv 6447 (class class class)co 7295 0cc0 10899 1c1 10900 ℕcn 12001 ℕ0cn0 12261 ℤ≥cuz 12610 ...cfz 13267 seqcseq 13749 Basecbs 16940 +gcplusg 16990 0gc0g 17178 Mndcmnd 18413 .gcmg 18728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-n0 12262 df-z 12348 df-uz 12611 df-fz 13268 df-seq 13750 df-0g 17180 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-mulg 18729 |
This theorem is referenced by: mulgz 18759 mulgnn0ass 18767 odmodnn0 19176 mulgmhm 19457 srg1expzeq1 19803 lply1binomsc 21506 tsmsxp 23334 frobrhm 31513 |
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