Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0repnf | Structured version Visualization version GIF version |
Description: The of nonnegative extended reals is a real number if and only if it is not +∞. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0repnf.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
sge0repnf.f | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
Ref | Expression |
---|---|
sge0repnf | ⊢ (𝜑 → ((Σ^‘𝐹) ∈ ℝ ↔ ¬ (Σ^‘𝐹) = +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renepnf 10864 | . . . 4 ⊢ ((Σ^‘𝐹) ∈ ℝ → (Σ^‘𝐹) ≠ +∞) | |
2 | 1 | neneqd 2940 | . . 3 ⊢ ((Σ^‘𝐹) ∈ ℝ → ¬ (Σ^‘𝐹) = +∞) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ ℝ → ¬ (Σ^‘𝐹) = +∞)) |
4 | rge0ssre 13027 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ | |
5 | 0xr 10863 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → 0 ∈ ℝ*) |
7 | pnfxr 10870 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
8 | 7 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → +∞ ∈ ℝ*) |
9 | sge0repnf.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
10 | sge0repnf.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) | |
11 | 9, 10 | sge0xrcl 43552 | . . . . . 6 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*) |
12 | 11 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ∈ ℝ*) |
13 | 9, 10 | sge0ge0 43551 | . . . . . 6 ⊢ (𝜑 → 0 ≤ (Σ^‘𝐹)) |
14 | 13 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → 0 ≤ (Σ^‘𝐹)) |
15 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → ¬ (Σ^‘𝐹) = +∞) | |
16 | nltpnft 12737 | . . . . . . . . 9 ⊢ ((Σ^‘𝐹) ∈ ℝ* → ((Σ^‘𝐹) = +∞ ↔ ¬ (Σ^‘𝐹) < +∞)) | |
17 | 11, 16 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((Σ^‘𝐹) = +∞ ↔ ¬ (Σ^‘𝐹) < +∞)) |
18 | 17 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → ((Σ^‘𝐹) = +∞ ↔ ¬ (Σ^‘𝐹) < +∞)) |
19 | 15, 18 | mtbid 327 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → ¬ ¬ (Σ^‘𝐹) < +∞) |
20 | 19 | notnotrd 135 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) < +∞) |
21 | 6, 8, 12, 14, 20 | elicod 12968 | . . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ∈ (0[,)+∞)) |
22 | 4, 21 | sseldi 3889 | . . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ∈ ℝ) |
23 | 22 | ex 416 | . 2 ⊢ (𝜑 → (¬ (Σ^‘𝐹) = +∞ → (Σ^‘𝐹) ∈ ℝ)) |
24 | 3, 23 | impbid 215 | 1 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ ℝ ↔ ¬ (Σ^‘𝐹) = +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5043 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ℝcr 10711 0cc0 10712 +∞cpnf 10847 ℝ*cxr 10849 < clt 10850 ≤ cle 10851 [,)cico 12920 [,]cicc 12921 Σ^csumge0 43529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-oi 9115 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-sum 15233 df-sumge0 43530 |
This theorem is referenced by: sge0rern 43555 sge0supre 43556 sge0less 43559 sge0le 43574 sge0split 43576 sge0iunmpt 43585 sge0rpcpnf 43588 sge0xadd 43602 sge0repnfmpt 43606 sge0gtfsumgt 43610 omeiunltfirp 43686 hoidmv1lelem1 43758 hoidmv1lelem2 43759 hoidmv1lelem3 43760 hoidmv1le 43761 hoidmvlelem3 43764 hoidmvlelem5 43766 |
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