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Mirrors > Home > ILE Home > Th. List > axcaucvglemf | GIF version |
Description: Lemma for axcaucvg 7912. Mapping to N and R yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.) |
Ref | Expression |
---|---|
axcaucvg.n | β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} |
axcaucvg.f | β’ (π β πΉ:πβΆβ) |
axcaucvg.cau | β’ (π β βπ β π βπ β π (π <β π β ((πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1)) β§ (πΉβπ) <β ((πΉβπ) + (β©π β β (π Β· π) = 1))))) |
axcaucvg.g | β’ πΊ = (π β N β¦ (β©π§ β R (πΉββ¨[β¨(β¨{π β£ π <Q [β¨π, 1oβ©] ~Q }, {π’ β£ [β¨π, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) = β¨π§, 0Rβ©)) |
Ref | Expression |
---|---|
axcaucvglemf | β’ (π β πΊ:NβΆR) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axcaucvg.n | . . 3 β’ π = β© {π₯ β£ (1 β π₯ β§ βπ¦ β π₯ (π¦ + 1) β π₯)} | |
2 | axcaucvg.f | . . 3 β’ (π β πΉ:πβΆβ) | |
3 | 1, 2 | axcaucvglemcl 7907 | . 2 β’ ((π β§ π β N) β (β©π§ β R (πΉββ¨[β¨(β¨{π β£ π <Q [β¨π, 1oβ©] ~Q }, {π’ β£ [β¨π, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) = β¨π§, 0Rβ©) β R) |
4 | axcaucvg.g | . 2 β’ πΊ = (π β N β¦ (β©π§ β R (πΉββ¨[β¨(β¨{π β£ π <Q [β¨π, 1oβ©] ~Q }, {π’ β£ [β¨π, 1oβ©] ~Q <Q π’}β© +P 1P), 1Pβ©] ~R , 0Rβ©) = β¨π§, 0Rβ©)) | |
5 | 3, 4 | fmptd 5683 | 1 β’ (π β πΊ:NβΆR) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1363 β wcel 2158 {cab 2173 βwral 2465 β¨cop 3607 β© cint 3856 class class class wbr 4015 β¦ cmpt 4076 βΆwf 5224 βcfv 5228 β©crio 5843 (class class class)co 5888 1oc1o 6423 [cec 6546 Ncnpi 7284 ~Q ceq 7291 <Q cltq 7297 1Pc1p 7304 +P cpp 7305 ~R cer 7308 Rcnr 7309 0Rc0r 7310 βcr 7823 1c1 7825 + caddc 7827 <β cltrr 7828 Β· cmul 7829 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-eprel 4301 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-1o 6430 df-2o 6431 df-oadd 6434 df-omul 6435 df-er 6548 df-ec 6550 df-qs 6554 df-ni 7316 df-pli 7317 df-mi 7318 df-lti 7319 df-plpq 7356 df-mpq 7357 df-enq 7359 df-nqqs 7360 df-plqqs 7361 df-mqqs 7362 df-1nqqs 7363 df-rq 7364 df-ltnqqs 7365 df-enq0 7436 df-nq0 7437 df-0nq0 7438 df-plq0 7439 df-mq0 7440 df-inp 7478 df-i1p 7479 df-iplp 7480 df-enr 7738 df-nr 7739 df-plr 7740 df-0r 7743 df-1r 7744 df-c 7830 df-1 7832 df-r 7834 df-add 7835 |
This theorem is referenced by: axcaucvglemcau 7910 axcaucvglemres 7911 |
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