Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > hashf2 | Structured version Visualization version GIF version |
Description: Lemma for hasheuni 32457. (Contributed by Thierry Arnoux, 19-Nov-2016.) |
Ref | Expression |
---|---|
hashf2 | ⊢ ♯:V⟶(0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashf 14165 | . 2 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
2 | nn0z 12456 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
3 | zre 12436 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
4 | rexr 11134 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
5 | 2, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ*) |
6 | nn0ge0 12371 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥) | |
7 | elxrge0 13302 | . . . . 5 ⊢ (𝑥 ∈ (0[,]+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥)) | |
8 | 5, 6, 7 | sylanbrc 583 | . . . 4 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ (0[,]+∞)) |
9 | 8 | ssriv 3946 | . . 3 ⊢ ℕ0 ⊆ (0[,]+∞) |
10 | 0xr 11135 | . . . . 5 ⊢ 0 ∈ ℝ* | |
11 | pnfxr 11142 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
12 | 0lepnf 12981 | . . . . 5 ⊢ 0 ≤ +∞ | |
13 | ubicc2 13310 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞)) | |
14 | 10, 11, 12, 13 | mp3an 1461 | . . . 4 ⊢ +∞ ∈ (0[,]+∞) |
15 | snssi 4766 | . . . 4 ⊢ (+∞ ∈ (0[,]+∞) → {+∞} ⊆ (0[,]+∞)) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ {+∞} ⊆ (0[,]+∞) |
17 | 9, 16 | unssi 4143 | . 2 ⊢ (ℕ0 ∪ {+∞}) ⊆ (0[,]+∞) |
18 | fss 6680 | . 2 ⊢ ((♯:V⟶(ℕ0 ∪ {+∞}) ∧ (ℕ0 ∪ {+∞}) ⊆ (0[,]+∞)) → ♯:V⟶(0[,]+∞)) | |
19 | 1, 17, 18 | mp2an 690 | 1 ⊢ ♯:V⟶(0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3443 ∪ cun 3906 ⊆ wss 3908 {csn 4584 class class class wbr 5103 ⟶wf 6487 (class class class)co 7349 ℝcr 10983 0cc0 10984 +∞cpnf 11119 ℝ*cxr 11121 ≤ cle 11123 ℕ0cn0 12346 ℤcz 12432 [,]cicc 13195 ♯chash 14157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-card 9808 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-n0 12347 df-xnn0 12419 df-z 12433 df-uz 12696 df-icc 13199 df-hash 14158 |
This theorem is referenced by: hasheuni 32457 cntmeas 32598 |
Copyright terms: Public domain | W3C validator |