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Mirrors > Home > ILE Home > Th. List > fzfigd | GIF version |
Description: Deduction form of fzfig 10206. (Contributed by Jim Kingdon, 21-May-2020.) |
Ref | Expression |
---|---|
fzfigd.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
fzfigd.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Ref | Expression |
---|---|
fzfigd | ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfigd.m | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | fzfigd.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | fzfig 10206 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ Fin) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 (class class class)co 5774 Fincfn 6634 ℤcz 9057 ...cfz 9793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-1o 6313 df-er 6429 df-en 6635 df-fin 6637 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-n0 8981 df-z 9058 df-uz 9330 df-fz 9794 |
This theorem is referenced by: iseqf1olemqf1o 10269 iseqf1olemjpcl 10271 iseqf1olemqpcl 10272 iseqf1olemfvp 10273 seq3f1olemqsum 10276 seq3f1olemstep 10277 seq3f1olemp 10278 fseq1hash 10550 hashfz 10570 fnfz0hash 10578 nnf1o 11148 summodclem2a 11153 summodclem2 11154 summodc 11155 zsumdc 11156 fsum3 11159 fisumss 11164 fsumm1 11188 fsum1p 11190 fisum0diag 11213 fsumrev 11215 fsumshft 11216 fisum0diag2 11219 iserabs 11247 binomlem 11255 binom1dif 11259 isumsplit 11263 arisum2 11271 pwm1geoserap1 11280 geo2sum 11286 cvgratnnlemabsle 11299 cvgratnnlemrate 11302 mertenslemub 11306 mertenslemi1 11307 mertenslem2 11308 mertensabs 11309 prodmodclem3 11347 prodmodclem2a 11348 prodmodclem2 11349 efcvgfsum 11376 efaddlem 11383 eirraplem 11486 phivalfi 11891 phicl2 11893 hashdvds 11900 phiprmpw 11901 cvgcmp2nlemabs 13230 trilpolemlt1 13237 |
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