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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 2735 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | mulg1.m | . . . 4 ⊢ · = (.g‘𝐺) | |
5 | eqid 2735 | . . . 4 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
6 | 2, 3, 4, 5 | mulgnn 19106 | . . 3 ⊢ ((1 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘1)) |
7 | 1, 6 | mpan 690 | . 2 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘1)) |
8 | 1z 12645 | . . 3 ⊢ 1 ∈ ℤ | |
9 | fvconst2g 7222 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 1 ∈ ℕ) → ((ℕ × {𝑋})‘1) = 𝑋) | |
10 | 1, 9 | mpan2 691 | . . 3 ⊢ (𝑋 ∈ 𝐵 → ((ℕ × {𝑋})‘1) = 𝑋) |
11 | 8, 10 | seq1i 14053 | . 2 ⊢ (𝑋 ∈ 𝐵 → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘1) = 𝑋) |
12 | 7, 11 | eqtrd 2775 | 1 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {csn 4631 × cxp 5687 ‘cfv 6563 (class class class)co 7431 1c1 11154 ℕcn 12264 seqcseq 14039 Basecbs 17245 +gcplusg 17298 .gcmg 19098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-mulg 19099 |
This theorem is referenced by: mulg2 19114 mulgnn0p1 19116 mulgm1 19125 mulgp1 19138 mulgnnass 19140 cycsubmcl 19232 cycsubggend 19236 cycsubgcl 19237 odm1inv 19586 odbezout 19591 od1 19592 odeq1 19593 gex1 19624 gsumsnfd 19984 ablfacrp 20101 pgpfac1lem2 20110 pgpfac1lem3 20112 ablsimpgfindlem1 20142 srgbinom 20249 mulgrhm 21506 zlmlmod 21555 frgpcyg 21610 freshmansdream 21611 evlslem1 22124 m2detleiblem5 22647 cayhamlem1 22888 cpmadugsumlemB 22896 ply1remlem 26219 fta1blem 26225 xrsmulgzz 32994 omndmul2 33072 isarchi3 33177 archirngz 33179 archiabllem1a 33181 elrgspnlem2 33233 elrgspnlem3 33234 ofldchr 33324 evl1deg1 33581 evl1deg2 33582 evl1deg3 33583 coe1vr1 33593 deg1vr 33594 primrootscoprbij 42084 aks6d1c1p8 42097 ringexp0nn 42116 aks6d1c5lem3 42119 aks6d1c6lem1 42152 ply1vr1smo 48228 |
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