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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 11171 | . 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 17131 | . . . . 5 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
6 | eqid 2610 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | mulgnn0z.t | . . . . . 6 ⊢ · = (.g‘𝐺) | |
8 | eqid 2610 | . . . . . 6 ⊢ seq1((+g‘𝐺), (ℕ × { 0 })) = seq1((+g‘𝐺), (ℕ × { 0 })) | |
9 | 3, 6, 7, 8 | mulgnn 17370 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 0 ∈ 𝐵) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) |
10 | 2, 5, 9 | syl2anr 494 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (𝑁 · 0 ) = (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁)) |
11 | 3, 6, 4 | mndlid 17134 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
12 | 5, 11 | mpdan 699 | . . . . . 6 ⊢ (𝐺 ∈ Mnd → ( 0 (+g‘𝐺) 0 ) = 0 ) |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → ( 0 (+g‘𝐺) 0 ) = 0 ) |
14 | simpr 476 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
15 | nnuz 11599 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
16 | 14, 15 | syl6eleq 2698 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (ℤ≥‘1)) |
17 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → 0 ∈ 𝐵) |
18 | elfznn 12241 | . . . . . 6 ⊢ (𝑥 ∈ (1...𝑁) → 𝑥 ∈ ℕ) | |
19 | fvconst2g 6372 | . . . . . 6 ⊢ (( 0 ∈ 𝐵 ∧ 𝑥 ∈ ℕ) → ((ℕ × { 0 })‘𝑥) = 0 ) | |
20 | 17, 18, 19 | syl2an 493 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (1...𝑁)) → ((ℕ × { 0 })‘𝑥) = 0 ) |
21 | 13, 16, 20 | seqid3 12707 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (seq1((+g‘𝐺), (ℕ × { 0 }))‘𝑁) = 0 ) |
22 | 10, 21 | eqtrd 2644 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ) → (𝑁 · 0 ) = 0 ) |
23 | oveq1 6556 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 · 0 ) = (0 · 0 )) | |
24 | 3, 4, 7 | mulg0 17369 | . . . . 5 ⊢ ( 0 ∈ 𝐵 → (0 · 0 ) = 0 ) |
25 | 5, 24 | syl 17 | . . . 4 ⊢ (𝐺 ∈ Mnd → (0 · 0 ) = 0 ) |
26 | 23, 25 | sylan9eqr 2666 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 = 0) → (𝑁 · 0 ) = 0 ) |
27 | 22, 26 | jaodan 822 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝑁 · 0 ) = 0 ) |
28 | 1, 27 | sylan2b 491 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0) → (𝑁 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 × cxp 5036 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ℕcn 10897 ℕ0cn0 11169 ℤ≥cuz 11563 ...cfz 12197 seqcseq 12663 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Mndcmnd 17117 .gcmg 17363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-seq 12664 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mulg 17364 |
This theorem is referenced by: mulgz 17391 mulgnn0ass 17401 odmodnn0 17782 mulgmhm 18056 srg1expzeq1 18362 lply1binomsc 19498 tsmsxp 21768 |
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