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Mirrors > Home > ILE Home > Th. List > mulg0 | 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 9260 | . 2 ⊢ 0 ∈ ℤ | |
2 | mulg0.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2177 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | mulg0.o | . . . 4 ⊢ 0 = (0g‘𝐺) | |
5 | eqid 2177 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
6 | mulg0.t | . . . 4 ⊢ · = (.g‘𝐺) | |
7 | eqid 2177 | . . . 4 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
8 | 2, 3, 4, 5, 6, 7 | mulgval 12918 | . . 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 2177 | . . . 4 ⊢ 0 = 0 | |
10 | 9 | iftruei 3540 | . . 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 | eqtrdi 2226 | . 2 ⊢ ((0 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = 0 ) |
12 | 1, 11 | mpan 424 | 1 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ifcif 3534 {csn 3592 class class class wbr 4002 × cxp 4623 ‘cfv 5215 (class class class)co 5872 0cc0 7808 1c1 7809 < clt 7988 -cneg 8125 ℕcn 8915 ℤcz 9249 seqcseq 10440 Basecbs 12454 +gcplusg 12528 0gc0g 12693 invgcminusg 12810 .gcmg 12915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-frec 6389 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-inn 8916 df-2 8974 df-n0 9173 df-z 9250 df-uz 9525 df-seqfrec 10441 df-ndx 12457 df-slot 12458 df-base 12460 df-plusg 12541 df-0g 12695 df-minusg 12813 df-mulg 12916 |
This theorem is referenced by: mulgnn0p1 12926 mulgnn0subcl 12928 mulgneg 12933 mulgaddcom 12938 mulginvcom 12939 mulgnn0z 12941 mulgnn0dir 12944 mulgneg2 12948 mulgnn0ass 12950 mhmmulg 12955 srgmulgass 13103 srgpcomp 13104 mulgass2 13166 cnfldmulg 13339 cnfldexp 13340 |
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