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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 11292 | . . . 4 ⊢ ((Σ^‘𝐹) ∈ ℝ → (Σ^‘𝐹) ≠ +∞) | |
2 | 1 | neneqd 2935 | . . 3 ⊢ ((Σ^‘𝐹) ∈ ℝ → ¬ (Σ^‘𝐹) = +∞) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ ℝ → ¬ (Σ^‘𝐹) = +∞)) |
4 | rge0ssre 13465 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ | |
5 | 0xr 11291 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → 0 ∈ ℝ*) |
7 | pnfxr 11298 | . . . . . 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 45836 | . . . . . 6 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*) |
12 | 11 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ∈ ℝ*) |
13 | 9, 10 | sge0ge0 45835 | . . . . . 6 ⊢ (𝜑 → 0 ≤ (Σ^‘𝐹)) |
14 | 13 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → 0 ≤ (Σ^‘𝐹)) |
15 | simpr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → ¬ (Σ^‘𝐹) = +∞) | |
16 | nltpnft 13175 | . . . . . . . . 9 ⊢ ((Σ^‘𝐹) ∈ ℝ* → ((Σ^‘𝐹) = +∞ ↔ ¬ (Σ^‘𝐹) < +∞)) | |
17 | 11, 16 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ((Σ^‘𝐹) = +∞ ↔ ¬ (Σ^‘𝐹) < +∞)) |
18 | 17 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → ((Σ^‘𝐹) = +∞ ↔ ¬ (Σ^‘𝐹) < +∞)) |
19 | 15, 18 | mtbid 323 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → ¬ ¬ (Σ^‘𝐹) < +∞) |
20 | 19 | notnotrd 133 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) < +∞) |
21 | 6, 8, 12, 14, 20 | elicod 13406 | . . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ∈ (0[,)+∞)) |
22 | 4, 21 | sselid 3970 | . . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐹) = +∞) → (Σ^‘𝐹) ∈ ℝ) |
23 | 22 | ex 411 | . 2 ⊢ (𝜑 → (¬ (Σ^‘𝐹) = +∞ → (Σ^‘𝐹) ∈ ℝ)) |
24 | 3, 23 | impbid 211 | 1 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ ℝ ↔ ¬ (Σ^‘𝐹) = +∞)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5143 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 ℝcr 11137 0cc0 11138 +∞cpnf 11275 ℝ*cxr 11277 < clt 11278 ≤ cle 11279 [,)cico 13358 [,]cicc 13359 Σ^csumge0 45813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 df-sumge0 45814 |
This theorem is referenced by: sge0rern 45839 sge0supre 45840 sge0less 45843 sge0le 45858 sge0split 45860 sge0iunmpt 45869 sge0rpcpnf 45872 sge0xadd 45886 sge0repnfmpt 45890 sge0gtfsumgt 45894 omeiunltfirp 45970 hoidmv1lelem1 46042 hoidmv1lelem2 46043 hoidmv1lelem3 46044 hoidmv1le 46045 hoidmvlelem3 46048 hoidmvlelem5 46050 |
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