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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0rernmpt | Structured version Visualization version GIF version |
Description: If the sum of nonnegative extended reals is not +∞ then no term is +∞. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
sge0rernmpt.xph | ⊢ Ⅎ𝑥𝜑 |
sge0rernmpt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0rernmpt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
sge0rernmpt.re | ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ) |
Ref | Expression |
---|---|
sge0rernmpt | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 11207 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ*) |
3 | pnfxr 11214 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → +∞ ∈ ℝ*) |
5 | iccssxr 13353 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
6 | sge0rernmpt.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
7 | 5, 6 | sselid 3943 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
8 | iccgelb 13326 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
9 | 2, 4, 6, 8 | syl3anc 1372 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) |
10 | simpr 486 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → ¬ 𝐵 < +∞) | |
11 | nltpnft 13089 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ* → (𝐵 = +∞ ↔ ¬ 𝐵 < +∞)) | |
12 | 7, 11 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = +∞ ↔ ¬ 𝐵 < +∞)) |
13 | 12 | adantr 482 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → (𝐵 = +∞ ↔ ¬ 𝐵 < +∞)) |
14 | 10, 13 | mpbird 257 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → 𝐵 = +∞) |
15 | 14 | eqcomd 2739 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → +∞ = 𝐵) |
16 | simpr 486 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
17 | eqid 2733 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
18 | 17 | elrnmpt1 5914 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (0[,]+∞)) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
19 | 16, 6, 18 | syl2anc 585 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
20 | 19 | adantr 482 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
21 | 15, 20 | eqeltrd 2834 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → +∞ ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
22 | sge0rernmpt.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
23 | sge0rernmpt.xph | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
24 | 23, 6, 17 | fmptdf 7066 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
25 | sge0rernmpt.re | . . . . 5 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ) | |
26 | 22, 24, 25 | sge0rern 44715 | . . . 4 ⊢ (𝜑 → ¬ +∞ ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
27 | 26 | ad2antrr 725 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → ¬ +∞ ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
28 | 21, 27 | condan 817 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 < +∞) |
29 | 2, 4, 7, 9, 28 | elicod 13320 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 class class class wbr 5106 ↦ cmpt 5189 ran crn 5635 ‘cfv 6497 (class class class)co 7358 ℝcr 11055 0cc0 11056 +∞cpnf 11191 ℝ*cxr 11193 < clt 11194 ≤ cle 11195 [,)cico 13272 [,]cicc 13273 Σ^csumge0 44689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 df-sumge0 44690 |
This theorem is referenced by: sge0ltfirpmpt2 44753 sge0xadd 44762 |
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