hashf2 - Mathbox for Thierry Arnoux

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Theorem hashf2 33697

Description: Lemma for hasheuni 33698. (Contributed by Thierry Arnoux, 19-Nov-2016.)

**Assertion**
Ref	Expression
hashf2	⊢ ♯:V⟶(0[,]+∞)

**Proof of Theorem hashf2**
Step	Hyp	Ref	Expression
1		hashf 14323	. 2 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
2		nn0z 12607	. . . . . 6 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ∈ ℤ)
3		zre 12586	. . . . . 6 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ)
4		rexr 11284	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
5	2, 3, 4	3syl 18	. . . . 5 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ∈ ℝ^*)
6		nn0ge0 12521	. . . . 5 ⊢ (𝑥 ∈ ℕ₀ → 0 ≤ 𝑥)
7		elxrge0 13460	. . . . 5 ⊢ (𝑥 ∈ (0[,]+∞) ↔ (𝑥 ∈ ℝ^* ∧ 0 ≤ 𝑥))
8	5, 6, 7	sylanbrc 582	. . . 4 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ∈ (0[,]+∞))
9	8	ssriv 3982	. . 3 ⊢ ℕ₀ ⊆ (0[,]+∞)
10		0xr 11285	. . . . 5 ⊢ 0 ∈ ℝ^*
11		pnfxr 11292	. . . . 5 ⊢ +∞ ∈ ℝ^*
12		0lepnf 13138	. . . . 5 ⊢ 0 ≤ +∞
13		ubicc2 13468	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞))
14	10, 11, 12, 13	mp3an 1458	. . . 4 ⊢ +∞ ∈ (0[,]+∞)
15		snssi 4807	. . . 4 ⊢ (+∞ ∈ (0[,]+∞) → {+∞} ⊆ (0[,]+∞))
16	14, 15	ax-mp 5	. . 3 ⊢ {+∞} ⊆ (0[,]+∞)
17	9, 16	unssi 4181	. 2 ⊢ (ℕ₀ ∪ {+∞}) ⊆ (0[,]+∞)
18		fss 6733	. 2 ⊢ ((♯:V⟶(ℕ₀ ∪ {+∞}) ∧ (ℕ₀ ∪ {+∞}) ⊆ (0[,]+∞)) → ♯:V⟶(0[,]+∞))
19	1, 17, 18	mp2an 691	1 ⊢ ♯:V⟶(0[,]+∞)

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2099 Vcvv 3470 ∪ cun 3943 ⊆ wss 3945 {csn 4624 class class class wbr 5142 ⟶wf 6538 (class class class)co 7414 ℝcr 11131 0cc0 11132 +∞cpnf 11269 ℝ^*cxr 11271 ≤ cle 11273 ℕ₀cn0 12496 ℤcz 12582 [,]cicc 13353 ♯chash 14315

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-xnn0 12569 df-z 12583 df-uz 12847 df-icc 13357 df-hash 14316

This theorem is referenced by: hasheuni 33698 cntmeas 33839

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