Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > mulg1 | GIF version | ||
| Description: Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.) |
| Ref | Expression |
|---|---|
| mulg1.b | β’ π΅ = (BaseβπΊ) |
| mulg1.m | β’ Β· = (.gβπΊ) |
| Ref | Expression |
|---|---|
| mulg1 | β’ (π β π΅ β (1 Β· π) = π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1nn 8932 | . . 3 β’ 1 β β | |
| 2 | mulg1.b | . . . 4 β’ π΅ = (BaseβπΊ) | |
| 3 | eqid 2177 | . . . 4 β’ (+gβπΊ) = (+gβπΊ) | |
| 4 | mulg1.m | . . . 4 β’ Β· = (.gβπΊ) | |
| 5 | eqid 2177 | . . . 4 β’ seq1((+gβπΊ), (β Γ {π})) = seq1((+gβπΊ), (β Γ {π})) | |
| 6 | 2, 3, 4, 5 | mulgnn 12994 | . . 3 β’ ((1 β β β§ π β π΅) β (1 Β· π) = (seq1((+gβπΊ), (β Γ {π}))β1)) |
| 7 | 1, 6 | mpan 424 | . 2 β’ (π β π΅ β (1 Β· π) = (seq1((+gβπΊ), (β Γ {π}))β1)) |
| 8 | 1zzd 9282 | . . 3 β’ (π β π΅ β 1 β β€) | |
| 9 | elnnuz 9566 | . . . 4 β’ (π’ β β β π’ β (β€β₯β1)) | |
| 10 | fvconst2g 5732 | . . . . . 6 β’ ((π β π΅ β§ π’ β β) β ((β Γ {π})βπ’) = π) | |
| 11 | simpl 109 | . . . . . 6 β’ ((π β π΅ β§ π’ β β) β π β π΅) | |
| 12 | 10, 11 | eqeltrd 2254 | . . . . 5 β’ ((π β π΅ β§ π’ β β) β ((β Γ {π})βπ’) β π΅) |
| 13 | 12 | elexd 2752 | . . . 4 β’ ((π β π΅ β§ π’ β β) β ((β Γ {π})βπ’) β V) |
| 14 | 9, 13 | sylan2br 288 | . . 3 β’ ((π β π΅ β§ π’ β (β€β₯β1)) β ((β Γ {π})βπ’) β V) |
| 15 | simprl 529 | . . . 4 β’ ((π β π΅ β§ (π’ β V β§ π£ β V)) β π’ β V) | |
| 16 | 2 | basmex 12523 | . . . . . 6 β’ (π β π΅ β πΊ β V) |
| 17 | plusgslid 12573 | . . . . . . 7 β’ (+g = Slot (+gβndx) β§ (+gβndx) β β) | |
| 18 | 17 | slotex 12491 | . . . . . 6 β’ (πΊ β V β (+gβπΊ) β V) |
| 19 | 16, 18 | syl 14 | . . . . 5 β’ (π β π΅ β (+gβπΊ) β V) |
| 20 | 19 | adantr 276 | . . . 4 β’ ((π β π΅ β§ (π’ β V β§ π£ β V)) β (+gβπΊ) β V) |
| 21 | simprr 531 | . . . 4 β’ ((π β π΅ β§ (π’ β V β§ π£ β V)) β π£ β V) | |
| 22 | ovexg 5911 | . . . 4 β’ ((π’ β V β§ (+gβπΊ) β V β§ π£ β V) β (π’(+gβπΊ)π£) β V) | |
| 23 | 15, 20, 21, 22 | syl3anc 1238 | . . 3 β’ ((π β π΅ β§ (π’ β V β§ π£ β V)) β (π’(+gβπΊ)π£) β V) |
| 24 | 8, 14, 23 | seq3-1 10462 | . 2 β’ (π β π΅ β (seq1((+gβπΊ), (β Γ {π}))β1) = ((β Γ {π})β1)) |
| 25 | fvconst2g 5732 | . . 3 β’ ((π β π΅ β§ 1 β β) β ((β Γ {π})β1) = π) | |
| 26 | 1, 25 | mpan2 425 | . 2 β’ (π β π΅ β ((β Γ {π})β1) = π) |
| 27 | 7, 24, 26 | 3eqtrd 2214 | 1 β’ (π β π΅ β (1 Β· π) = π) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 Vcvv 2739 {csn 3594 Γ cxp 4626 βcfv 5218 (class class class)co 5877 1c1 7814 βcn 8921 β€β₯cuz 9530 seqcseq 10447 Basecbs 12464 +gcplusg 12538 .gcmg 12988 |
| 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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-ltadd 7929 |
| 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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-inn 8922 df-2 8980 df-n0 9179 df-z 9256 df-uz 9531 df-seqfrec 10448 df-ndx 12467 df-slot 12468 df-base 12470 df-plusg 12551 df-0g 12712 df-minusg 12886 df-mulg 12989 |
| This theorem is referenced by: mulg2 12997 mulgnn0p1 12999 mulgm1 13008 mulgp1 13021 mulgnnass 13023 |
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