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Mirrors > Home > MPE Home > Th. List > mulgnn0p1 | Structured version Visualization version GIF version |
Description: Group multiple (exponentiation) operation at a successor, extended to ℕ0. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulgnn0p1.b | ⊢ 𝐵 = (Base‘𝐺) |
mulgnn0p1.t | ⊢ · = (.g‘𝐺) |
mulgnn0p1.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
mulgnn0p1 | ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 479 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
2 | simpl3 1250 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ 𝐵) | |
3 | mulgnn0p1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
4 | mulgnn0p1.t | . . . 4 ⊢ · = (.g‘𝐺) | |
5 | mulgnn0p1.p | . . . 4 ⊢ + = (+g‘𝐺) | |
6 | 3, 4, 5 | mulgnnp1 17910 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
7 | 1, 2, 6 | syl2anc 579 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
8 | eqid 2825 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | 3, 5, 8 | mndlid 17671 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋) |
10 | 3, 8, 4 | mulg0 17907 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺)) |
11 | 10 | adantl 475 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = (0g‘𝐺)) |
12 | 11 | oveq1d 6925 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0 · 𝑋) + 𝑋) = ((0g‘𝐺) + 𝑋)) |
13 | 3, 4 | mulg1 17909 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
14 | 13 | adantl 475 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = 𝑋) |
15 | 9, 12, 14 | 3eqtr4rd 2872 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
16 | 15 | 3adant2 1165 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
17 | oveq1 6917 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
18 | 1e0p1 11871 | . . . . . . 7 ⊢ 1 = (0 + 1) | |
19 | 17, 18 | syl6eqr 2879 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1) |
20 | 19 | oveq1d 6925 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = (1 · 𝑋)) |
21 | oveq1 6917 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
22 | 21 | oveq1d 6925 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 · 𝑋) + 𝑋) = ((0 · 𝑋) + 𝑋)) |
23 | 20, 22 | eqeq12d 2840 | . . . 4 ⊢ (𝑁 = 0 → (((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋) ↔ (1 · 𝑋) = ((0 · 𝑋) + 𝑋))) |
24 | 16, 23 | syl5ibrcom 239 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))) |
25 | 24 | imp 397 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 = 0) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
26 | simp2 1171 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
27 | elnn0 11627 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
28 | 26, 27 | sylib 210 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
29 | 7, 25, 28 | mpjaodan 986 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 878 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 0cc0 10259 1c1 10260 + caddc 10262 ℕcn 11357 ℕ0cn0 11625 Basecbs 16229 +gcplusg 16312 0gc0g 16460 Mndcmnd 17654 .gcmg 17901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-seq 13103 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mulg 17902 |
This theorem is referenced by: mulgaddcom 17924 mulginvcom 17925 mulgneg2 17934 mhmmulg 17941 srgmulgass 18892 srgpcomp 18893 srgpcompp 18894 srgbinomlem4 18904 srgbinomlem 18905 lmodvsmmulgdi 19261 assamulgscmlem2 19717 mplcoe3 19834 cnfldmulg 20145 cnfldexp 20146 tmdmulg 22273 clmmulg 23277 omndmul 30255 lmodvsmdi 43024 |
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