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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 483 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
2 | simpl3 1190 | . . 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 19071 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
7 | 1, 2, 6 | syl2anc 582 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
8 | eqid 2726 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | 3, 5, 8 | mndlid 18741 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋) |
10 | 3, 8, 4 | mulg0 19063 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺)) |
11 | 10 | adantl 480 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = (0g‘𝐺)) |
12 | 11 | oveq1d 7430 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0 · 𝑋) + 𝑋) = ((0g‘𝐺) + 𝑋)) |
13 | 3, 4 | mulg1 19070 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
14 | 13 | adantl 480 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = 𝑋) |
15 | 9, 12, 14 | 3eqtr4rd 2777 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
16 | 15 | 3adant2 1128 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
17 | oveq1 7422 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
18 | 1e0p1 12764 | . . . . . . 7 ⊢ 1 = (0 + 1) | |
19 | 17, 18 | eqtr4di 2784 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1) |
20 | 19 | oveq1d 7430 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = (1 · 𝑋)) |
21 | oveq1 7422 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
22 | 21 | oveq1d 7430 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 · 𝑋) + 𝑋) = ((0 · 𝑋) + 𝑋)) |
23 | 20, 22 | eqeq12d 2742 | . . . 4 ⊢ (𝑁 = 0 → (((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋) ↔ (1 · 𝑋) = ((0 · 𝑋) + 𝑋))) |
24 | 16, 23 | syl5ibrcom 246 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))) |
25 | 24 | imp 405 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 = 0) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
26 | simp2 1134 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
27 | elnn0 12519 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
28 | 26, 27 | sylib 217 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
29 | 7, 25, 28 | mpjaodan 956 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6545 (class class class)co 7415 0cc0 11148 1c1 11149 + caddc 11151 ℕcn 12257 ℕ0cn0 12517 Basecbs 17207 +gcplusg 17260 0gc0g 17448 Mndcmnd 18721 .gcmg 19056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8966 df-dom 8967 df-sdom 8968 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12258 df-n0 12518 df-z 12604 df-uz 12868 df-seq 14015 df-0g 17450 df-mgm 18627 df-sgrp 18706 df-mnd 18722 df-mulg 19057 |
This theorem is referenced by: mulgaddcom 19087 mulginvcom 19088 mulgneg2 19097 mhmmulg 19104 srgmulgass 20195 srgpcomp 20196 srgpcompp 20197 srgbinomlem4 20207 srgbinomlem 20208 lmodvsmmulgdi 20868 cnfldmulg 21390 cnfldexp 21391 assamulgscmlem2 21892 mplcoe3 22040 mhppwdeg 22139 tmdmulg 24083 clmmulg 25115 omndmul 32952 rprmdvdspow 33413 primrootsunit1 41808 aks6d1c1p6 41825 idomnnzpownz 41843 deg1pow 41852 domnexpgn0cl 42212 abvexp 42221 lmodvsmdi 47796 |
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