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 10086 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ∈ ℝ*) |
3 | pnfxr 10092 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | 3 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → +∞ ∈ ℝ*) |
5 | iccssxr 12256 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
6 | sge0rernmpt.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
7 | 5, 6 | sseldi 3601 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
8 | iccgelb 12230 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
9 | 2, 4, 6, 8 | syl3anc 1326 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵) |
10 | simpr 477 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → ¬ 𝐵 < +∞) | |
11 | nltpnft 11995 | . . . . . . . 8 ⊢ (𝐵 ∈ ℝ* → (𝐵 = +∞ ↔ ¬ 𝐵 < +∞)) | |
12 | 7, 11 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = +∞ ↔ ¬ 𝐵 < +∞)) |
13 | 12 | adantr 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → (𝐵 = +∞ ↔ ¬ 𝐵 < +∞)) |
14 | 10, 13 | mpbird 247 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → 𝐵 = +∞) |
15 | 14 | eqcomd 2628 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → +∞ = 𝐵) |
16 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
17 | eqid 2622 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
18 | 17 | elrnmpt1 5374 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (0[,]+∞)) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
19 | 16, 6, 18 | syl2anc 693 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
20 | 19 | adantr 481 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → 𝐵 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
21 | 15, 20 | eqeltrd 2701 | . . 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 6387 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
25 | sge0rernmpt.re | . . . . 5 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ) | |
26 | 22, 24, 25 | sge0rern 40605 | . . . 4 ⊢ (𝜑 → ¬ +∞ ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
27 | 26 | ad2antrr 762 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝐵 < +∞) → ¬ +∞ ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) |
28 | 21, 27 | condan 835 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 < +∞) |
29 | 2, 4, 7, 9, 28 | elicod 12224 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 [,)cico 12177 [,]cicc 12178 Σ^csumge0 40579 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-sumge0 40580 |
This theorem is referenced by: sge0ltfirpmpt2 40643 sge0xadd 40652 |
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