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Mirrors > Home > MPE Home > Th. List > mulg0 | Structured version Visualization version GIF version |
Description: Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulg0.b | ⊢ 𝐵 = (Base‘𝐺) |
mulg0.o | ⊢ 0 = (0g‘𝐺) |
mulg0.t | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
mulg0 | ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11995 | . 2 ⊢ 0 ∈ ℤ | |
2 | mulg0.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2823 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | mulg0.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
5 | eqid 2823 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
6 | mulg0.t | . . . 4 ⊢ · = (.g‘𝐺) | |
7 | eqid 2823 | . . . 4 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
8 | 2, 3, 4, 5, 6, 7 | mulgval 18230 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = if(0 = 0, 0 , if(0 < 0, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘0), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-0))))) |
9 | eqid 2823 | . . . 4 ⊢ 0 = 0 | |
10 | 9 | iftruei 4476 | . . 3 ⊢ if(0 = 0, 0 , if(0 < 0, (seq1((+g‘𝐺), (ℕ × {𝑋}))‘0), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑋}))‘-0)))) = 0 |
11 | 8, 10 | syl6eq 2874 | . 2 ⊢ ((0 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = 0 ) |
12 | 1, 11 | mpan 688 | 1 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ifcif 4469 {csn 4569 class class class wbr 5068 × cxp 5555 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 < clt 10677 -cneg 10873 ℕcn 11640 ℤcz 11984 seqcseq 13372 Basecbs 16485 +gcplusg 16567 0gc0g 16715 invgcminusg 18106 .gcmg 18226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-seq 13373 df-mulg 18227 |
This theorem is referenced by: mulgnn0gsum 18236 mulgnn0p1 18241 mulgnn0subcl 18243 mulgneg 18248 mulgaddcom 18253 mulginvcom 18254 mulgnn0z 18256 mulgnn0dir 18259 mulgneg2 18263 mulgnn0ass 18265 mhmmulg 18270 submmulg 18273 cycsubm 18347 odid 18668 oddvdsnn0 18674 oddvds 18677 odf1 18691 gexid 18708 mulgnn0di 18948 0cyg 19015 gsumconst 19056 srgmulgass 19283 srgpcomp 19284 srgbinomlem3 19294 srgbinomlem4 19295 srgbinom 19297 mulgass2 19353 lmodvsmmulgdi 19671 assamulgscmlem1 20130 mplcoe3 20249 mplcoe5 20251 mplbas2 20253 psrbagev1 20292 evlslem3 20295 evlslem1 20297 ply1scltm 20451 cnfldmulg 20579 cnfldexp 20580 chfacfscmulgsum 21470 chfacfpmmulgsum 21474 cpmadugsumlemF 21486 tmdmulg 22702 clmmulg 23707 dchrptlem2 25843 xrsmulgzz 30667 ressmulgnn0 30673 omndmul2 30715 omndmul 30717 archirng 30819 archirngz 30820 archiabllem1b 30823 archiabllem2c 30826 freshmansdream 30861 lmodvsmdi 44437 |
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