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Mirrors > Home > MPE Home > Th. List > Mathboxes > hashf2 | Structured version Visualization version GIF version |
Description: Lemma for hasheuni 31346. (Contributed by Thierry Arnoux, 19-Nov-2016.) |
Ref | Expression |
---|---|
hashf2 | ⊢ ♯:V⟶(0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashf 13701 | . 2 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
2 | nn0z 12008 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℤ) | |
3 | zre 11988 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
4 | rexr 10689 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
5 | 2, 3, 4 | 3syl 18 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ*) |
6 | nn0ge0 11925 | . . . . 5 ⊢ (𝑥 ∈ ℕ0 → 0 ≤ 𝑥) | |
7 | elxrge0 12848 | . . . . 5 ⊢ (𝑥 ∈ (0[,]+∞) ↔ (𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥)) | |
8 | 5, 6, 7 | sylanbrc 585 | . . . 4 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ (0[,]+∞)) |
9 | 8 | ssriv 3973 | . . 3 ⊢ ℕ0 ⊆ (0[,]+∞) |
10 | 0xr 10690 | . . . . 5 ⊢ 0 ∈ ℝ* | |
11 | pnfxr 10697 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
12 | 0lepnf 12530 | . . . . 5 ⊢ 0 ≤ +∞ | |
13 | ubicc2 12856 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞)) | |
14 | 10, 11, 12, 13 | mp3an 1457 | . . . 4 ⊢ +∞ ∈ (0[,]+∞) |
15 | snssi 4743 | . . . 4 ⊢ (+∞ ∈ (0[,]+∞) → {+∞} ⊆ (0[,]+∞)) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ {+∞} ⊆ (0[,]+∞) |
17 | 9, 16 | unssi 4163 | . 2 ⊢ (ℕ0 ∪ {+∞}) ⊆ (0[,]+∞) |
18 | fss 6529 | . 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 2114 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 {csn 4569 class class class wbr 5068 ⟶wf 6353 (class class class)co 7158 ℝcr 10538 0cc0 10539 +∞cpnf 10674 ℝ*cxr 10676 ≤ cle 10678 ℕ0cn0 11900 ℤcz 11984 [,]cicc 12744 ♯chash 13693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-icc 12748 df-hash 13694 |
This theorem is referenced by: hasheuni 31346 cntmeas 31487 |
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