![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mulgnn | Structured version Visualization version GIF version |
Description: Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulgnn.b | ⊢ 𝐵 = (Base‘𝐺) |
mulgnn.p | ⊢ + = (+g‘𝐺) |
mulgnn.t | ⊢ · = (.g‘𝐺) |
mulgnn.s | ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) |
Ref | Expression |
---|---|
mulgnn | ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝑆‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 12632 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | mulgnn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | mulgnn.p | . . . 4 ⊢ + = (+g‘𝐺) | |
4 | eqid 2735 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | eqid 2735 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
6 | mulgnn.t | . . . 4 ⊢ · = (.g‘𝐺) | |
7 | mulgnn.s | . . . 4 ⊢ 𝑆 = seq1( + , (ℕ × {𝑋})) | |
8 | 2, 3, 4, 5, 6, 7 | mulgval 19102 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁))))) |
9 | 1, 8 | sylan 580 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁))))) |
10 | nnne0 12298 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
11 | 10 | neneqd 2943 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0) |
12 | 11 | iffalsed 4542 | . . . 4 ⊢ (𝑁 ∈ ℕ → if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁)))) = if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁)))) |
13 | nngt0 12295 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
14 | 13 | iftrued 4539 | . . . 4 ⊢ (𝑁 ∈ ℕ → if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁))) = (𝑆‘𝑁)) |
15 | 12, 14 | eqtrd 2775 | . . 3 ⊢ (𝑁 ∈ ℕ → if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁)))) = (𝑆‘𝑁)) |
16 | 15 | adantr 480 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → if(𝑁 = 0, (0g‘𝐺), if(0 < 𝑁, (𝑆‘𝑁), ((invg‘𝐺)‘(𝑆‘-𝑁)))) = (𝑆‘𝑁)) |
17 | 9, 16 | eqtrd 2775 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝑆‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ifcif 4531 {csn 4631 class class class wbr 5148 × cxp 5687 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 < clt 11293 -cneg 11491 ℕcn 12264 ℤcz 12611 seqcseq 14039 Basecbs 17245 +gcplusg 17298 0gc0g 17486 invgcminusg 18965 .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: ressmulgnn 19107 ressmulgnnd 19109 mulgnngsum 19110 mulg1 19112 mulgnnp1 19113 mulgnegnn 19115 mulgnnsubcl 19117 mulgnn0z 19132 mulgnndir 19134 submmulg 19149 subgmulg 19171 mulgnn0di 19858 gsumconst 19967 |
Copyright terms: Public domain | W3C validator |