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Mirrors > Home > MPE Home > Th. List > mulg1 | Structured version Visualization version GIF version |
Description: Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulg1.b | ⊢ 𝐵 = (Base‘𝐺) |
mulg1.m | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
mulg1 | ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 12275 | . . 3 ⊢ 1 ∈ ℕ | |
2 | mulg1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2726 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | mulg1.m | . . . 4 ⊢ · = (.g‘𝐺) | |
5 | eqid 2726 | . . . 4 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
6 | 2, 3, 4, 5 | mulgnn 19069 | . . 3 ⊢ ((1 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘1)) |
7 | 1, 6 | mpan 688 | . 2 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘1)) |
8 | 1z 12644 | . . 3 ⊢ 1 ∈ ℤ | |
9 | fvconst2g 7219 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 1 ∈ ℕ) → ((ℕ × {𝑋})‘1) = 𝑋) | |
10 | 1, 9 | mpan2 689 | . . 3 ⊢ (𝑋 ∈ 𝐵 → ((ℕ × {𝑋})‘1) = 𝑋) |
11 | 8, 10 | seq1i 14035 | . 2 ⊢ (𝑋 ∈ 𝐵 → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘1) = 𝑋) |
12 | 7, 11 | eqtrd 2766 | 1 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {csn 4633 × cxp 5680 ‘cfv 6554 (class class class)co 7424 1c1 11159 ℕcn 12264 seqcseq 14021 Basecbs 17213 +gcplusg 17266 .gcmg 19061 |
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 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
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-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-seq 14022 df-mulg 19062 |
This theorem is referenced by: mulg2 19077 mulgnn0p1 19079 mulgm1 19088 mulgp1 19101 mulgnnass 19103 cycsubmcl 19195 cycsubggend 19199 cycsubgcl 19200 odm1inv 19551 odbezout 19556 od1 19557 odeq1 19558 gex1 19589 gsumsnfd 19949 ablfacrp 20066 pgpfac1lem2 20075 pgpfac1lem3 20077 ablsimpgfindlem1 20107 srgbinom 20214 mulgrhm 21467 zlmlmod 21516 frgpcyg 21571 freshmansdream 21572 evlslem1 22097 m2detleiblem5 22618 cayhamlem1 22859 cpmadugsumlemB 22867 ply1remlem 26192 fta1blem 26198 xrsmulgzz 32889 omndmul2 32947 isarchi3 33052 archirngz 33054 archiabllem1a 33056 ofldchr 33192 evl1deg1 33448 evl1deg2 33449 evl1deg3 33450 coe1vr1 33460 deg1vr 33461 primrootscoprbij 41800 aks6d1c1p8 41813 ringexp0nn 41832 aks6d1c5lem3 41835 aks6d1c6lem1 41868 ply1vr1smo 47765 |
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