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| Mirrors > Home > MPE Home > Th. List > mulgfn | Structured version Visualization version GIF version | ||
| Description: Functionality of the group multiple operation. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| mulgfn.b | ⊢ 𝐵 = (Base‘𝐺) | 
| mulgfn.t | ⊢ · = (.g‘𝐺) | 
| Ref | Expression | 
|---|---|
| mulgfn | ⊢ · Fn (ℤ × 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mulgfn.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | eqid 2737 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 3 | eqid 2737 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 4 | eqid 2737 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
| 5 | mulgfn.t | . . 3 ⊢ · = (.g‘𝐺) | |
| 6 | 1, 2, 3, 4, 5 | mulgfval 19087 | . 2 ⊢ · = (𝑛 ∈ ℤ, 𝑥 ∈ 𝐵 ↦ if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛))))) | 
| 7 | fvex 6919 | . . 3 ⊢ (0g‘𝐺) ∈ V | |
| 8 | fvex 6919 | . . . 4 ⊢ (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛) ∈ V | |
| 9 | fvex 6919 | . . . 4 ⊢ ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛)) ∈ V | |
| 10 | 8, 9 | ifex 4576 | . . 3 ⊢ if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛))) ∈ V | 
| 11 | 7, 10 | ifex 4576 | . 2 ⊢ if(𝑛 = 0, (0g‘𝐺), if(0 < 𝑛, (seq1((+g‘𝐺), (ℕ × {𝑥}))‘𝑛), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑥}))‘-𝑛)))) ∈ V | 
| 12 | 6, 11 | fnmpoi 8095 | 1 ⊢ · Fn (ℤ × 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ifcif 4525 {csn 4626 class class class wbr 5143 × cxp 5683 Fn wfn 6556 ‘cfv 6561 0cc0 11155 1c1 11156 < clt 11295 -cneg 11493 ℕcn 12266 ℤcz 12613 seqcseq 14042 Basecbs 17247 +gcplusg 17297 0gc0g 17484 invgcminusg 18952 .gcmg 19085 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-mulg 19086 | 
| This theorem is referenced by: mulgfvi 19091 tgpmulg2 24102 | 
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