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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 487 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
2 | simpl3 1188 | . . 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 18228 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
7 | 1, 2, 6 | syl2anc 586 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
8 | eqid 2819 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | 3, 5, 8 | mndlid 17923 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋) |
10 | 3, 8, 4 | mulg0 18223 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺)) |
11 | 10 | adantl 484 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = (0g‘𝐺)) |
12 | 11 | oveq1d 7163 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0 · 𝑋) + 𝑋) = ((0g‘𝐺) + 𝑋)) |
13 | 3, 4 | mulg1 18227 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
14 | 13 | adantl 484 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = 𝑋) |
15 | 9, 12, 14 | 3eqtr4rd 2865 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
16 | 15 | 3adant2 1126 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
17 | oveq1 7155 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
18 | 1e0p1 12132 | . . . . . . 7 ⊢ 1 = (0 + 1) | |
19 | 17, 18 | syl6eqr 2872 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1) |
20 | 19 | oveq1d 7163 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = (1 · 𝑋)) |
21 | oveq1 7155 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
22 | 21 | oveq1d 7163 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 · 𝑋) + 𝑋) = ((0 · 𝑋) + 𝑋)) |
23 | 20, 22 | eqeq12d 2835 | . . . 4 ⊢ (𝑁 = 0 → (((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋) ↔ (1 · 𝑋) = ((0 · 𝑋) + 𝑋))) |
24 | 16, 23 | syl5ibrcom 249 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))) |
25 | 24 | imp 409 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 = 0) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
26 | simp2 1132 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
27 | elnn0 11891 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
28 | 26, 27 | sylib 220 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
29 | 7, 25, 28 | mpjaodan 955 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 0cc0 10529 1c1 10530 + caddc 10532 ℕcn 11630 ℕ0cn0 11889 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Mndcmnd 17903 .gcmg 18216 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-n0 11890 df-z 11974 df-uz 12236 df-seq 13362 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mulg 18217 |
This theorem is referenced by: mulgaddcom 18243 mulginvcom 18244 mulgneg2 18253 mhmmulg 18260 srgmulgass 19273 srgpcomp 19274 srgpcompp 19275 srgbinomlem4 19285 srgbinomlem 19286 lmodvsmmulgdi 19661 assamulgscmlem2 20121 mplcoe3 20239 cnfldmulg 20569 cnfldexp 20570 tmdmulg 22692 clmmulg 23697 omndmul 30708 lmodvsmdi 44421 |
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