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Mirrors > Home > MPE Home > Th. List > hashinf | Structured version Visualization version GIF version |
Description: The value of the # function on an infinite set. (Contributed by Mario Carneiro, 13-Jul-2014.) |
Ref | Expression |
---|---|
hashinf | ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | eldif 3550 | . . 3 ⊢ (𝐴 ∈ (V ∖ Fin) ↔ (𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin)) | |
3 | df-hash 12980 | . . . . . . 7 ⊢ # = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞})) | |
4 | 3 | reseq1i 5313 | . . . . . 6 ⊢ (# ↾ (V ∖ Fin)) = ((((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞})) ↾ (V ∖ Fin)) |
5 | resundir 5331 | . . . . . 6 ⊢ ((((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ∪ ((V ∖ Fin) × {+∞})) ↾ (V ∖ Fin)) = ((((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ↾ (V ∖ Fin)) ∪ (((V ∖ Fin) × {+∞}) ↾ (V ∖ Fin))) | |
6 | disjdif 3992 | . . . . . . . . 9 ⊢ (Fin ∩ (V ∖ Fin)) = ∅ | |
7 | eqid 2610 | . . . . . . . . . . 11 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) | |
8 | eqid 2610 | . . . . . . . . . . 11 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) | |
9 | 7, 8 | hashkf 12981 | . . . . . . . . . 10 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card):Fin⟶ℕ0 |
10 | ffn 5958 | . . . . . . . . . 10 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card):Fin⟶ℕ0 → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) Fn Fin) | |
11 | fnresdisj 5915 | . . . . . . . . . 10 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) Fn Fin → ((Fin ∩ (V ∖ Fin)) = ∅ ↔ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ↾ (V ∖ Fin)) = ∅)) | |
12 | 9, 10, 11 | mp2b 10 | . . . . . . . . 9 ⊢ ((Fin ∩ (V ∖ Fin)) = ∅ ↔ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ↾ (V ∖ Fin)) = ∅) |
13 | 6, 12 | mpbi 219 | . . . . . . . 8 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ↾ (V ∖ Fin)) = ∅ |
14 | pnfex 9972 | . . . . . . . . . 10 ⊢ +∞ ∈ V | |
15 | 14 | fconst 6004 | . . . . . . . . 9 ⊢ ((V ∖ Fin) × {+∞}):(V ∖ Fin)⟶{+∞} |
16 | ffn 5958 | . . . . . . . . 9 ⊢ (((V ∖ Fin) × {+∞}):(V ∖ Fin)⟶{+∞} → ((V ∖ Fin) × {+∞}) Fn (V ∖ Fin)) | |
17 | fnresdm 5914 | . . . . . . . . 9 ⊢ (((V ∖ Fin) × {+∞}) Fn (V ∖ Fin) → (((V ∖ Fin) × {+∞}) ↾ (V ∖ Fin)) = ((V ∖ Fin) × {+∞})) | |
18 | 15, 16, 17 | mp2b 10 | . . . . . . . 8 ⊢ (((V ∖ Fin) × {+∞}) ↾ (V ∖ Fin)) = ((V ∖ Fin) × {+∞}) |
19 | 13, 18 | uneq12i 3727 | . . . . . . 7 ⊢ ((((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ↾ (V ∖ Fin)) ∪ (((V ∖ Fin) × {+∞}) ↾ (V ∖ Fin))) = (∅ ∪ ((V ∖ Fin) × {+∞})) |
20 | uncom 3719 | . . . . . . 7 ⊢ (∅ ∪ ((V ∖ Fin) × {+∞})) = (((V ∖ Fin) × {+∞}) ∪ ∅) | |
21 | un0 3919 | . . . . . . 7 ⊢ (((V ∖ Fin) × {+∞}) ∪ ∅) = ((V ∖ Fin) × {+∞}) | |
22 | 19, 20, 21 | 3eqtri 2636 | . . . . . 6 ⊢ ((((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card) ↾ (V ∖ Fin)) ∪ (((V ∖ Fin) × {+∞}) ↾ (V ∖ Fin))) = ((V ∖ Fin) × {+∞}) |
23 | 4, 5, 22 | 3eqtri 2636 | . . . . 5 ⊢ (# ↾ (V ∖ Fin)) = ((V ∖ Fin) × {+∞}) |
24 | 23 | fveq1i 6104 | . . . 4 ⊢ ((# ↾ (V ∖ Fin))‘𝐴) = (((V ∖ Fin) × {+∞})‘𝐴) |
25 | fvres 6117 | . . . 4 ⊢ (𝐴 ∈ (V ∖ Fin) → ((# ↾ (V ∖ Fin))‘𝐴) = (#‘𝐴)) | |
26 | 14 | fvconst2 6374 | . . . 4 ⊢ (𝐴 ∈ (V ∖ Fin) → (((V ∖ Fin) × {+∞})‘𝐴) = +∞) |
27 | 24, 25, 26 | 3eqtr3a 2668 | . . 3 ⊢ (𝐴 ∈ (V ∖ Fin) → (#‘𝐴) = +∞) |
28 | 2, 27 | sylbir 224 | . 2 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) |
29 | 1, 28 | sylan 487 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ωcom 6957 reccrdg 7392 Fincfn 7841 cardccrd 8644 0cc0 9815 1c1 9816 + caddc 9818 +∞cpnf 9950 ℕ0cn0 11169 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-hash 12980 |
This theorem is referenced by: hashbnd 12985 hasheni 12998 hasheqf1oi 13002 hasheqf1oiOLD 13003 hashclb 13011 hasheq0 13015 hashdom 13029 hashdomi 13030 hashunx 13036 hashge1 13039 hashss 13058 hash1snb 13068 hashge2el2dif 13117 odhash 17812 lt6abl 18119 upgrfi 25758 umgrafi 25851 sizeusglecusg 26014 hashnbgravdg 26440 esumpinfsum 29466 hasheuni 29474 nfile 40369 pgrpgt2nabl 41941 |
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