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Mirrors > Home > ILE Home > Th. List > fz1f1o | GIF version |
Description: A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.) |
Ref | Expression |
---|---|
fz1f1o | ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 10632 | . . . 4 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
2 | elnn0 9071 | . . . 4 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) | |
3 | 1, 2 | sylib 121 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ∨ (♯‘𝐴) = 0)) |
4 | 3 | orcomd 719 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) ∈ ℕ)) |
5 | fihasheq0 10645 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅)) | |
6 | isfinite4im 10644 | . . . . 5 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴) | |
7 | bren 6681 | . . . . 5 ⊢ ((1...(♯‘𝐴)) ≈ 𝐴 ↔ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) | |
8 | 6, 7 | sylib 121 | . . . 4 ⊢ (𝐴 ∈ Fin → ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) |
9 | 8 | biantrud 302 | . . 3 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
10 | 5, 9 | orbi12d 783 | . 2 ⊢ (𝐴 ∈ Fin → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) ∈ ℕ) ↔ (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))) |
11 | 4, 10 | mpbid 146 | 1 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 698 = wceq 1332 ∃wex 1469 ∈ wcel 2125 ∅c0 3390 class class class wbr 3961 –1-1-onto→wf1o 5162 ‘cfv 5163 (class class class)co 5814 ≈ cen 6672 Fincfn 6674 0cc0 7711 1c1 7712 ℕcn 8812 ℕ0cn0 9069 ...cfz 9890 ♯chash 10626 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-addcom 7811 ax-addass 7813 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-0id 7819 ax-rnegex 7820 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-1st 6078 df-2nd 6079 df-recs 6242 df-frec 6328 df-1o 6353 df-er 6469 df-en 6675 df-dom 6676 df-fin 6677 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-inn 8813 df-n0 9070 df-z 9147 df-uz 9419 df-fz 9891 df-ihash 10627 |
This theorem is referenced by: isumz 11263 fsumf1o 11264 fisumss 11266 fsumcl2lem 11272 fsumadd 11280 fsummulc2 11322 prod1dc 11460 fprodf1o 11462 fprodssdc 11464 fprodmul 11465 |
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