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| Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpinfval | Structured version Visualization version GIF version | ||
| Description: The value of the extended sum of nonnegative terms, with at least one infinite term. (Contributed by Thierry Arnoux, 19-Jun-2017.) |
| Ref | Expression |
|---|---|
| esumpinfval.0 | ⊢ Ⅎ𝑘𝜑 |
| esumpinfval.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| esumpinfval.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
| esumpinfval.3 | ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞) |
| Ref | Expression |
|---|---|
| esumpinfval | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = +∞) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iccssxr 13470 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 2 | esumpinfval.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 3 | esumpinfval.0 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
| 4 | esumpinfval.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
| 5 | 4 | ex 412 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
| 6 | 3, 5 | ralrimi 3257 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
| 7 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
| 8 | 7 | esumcl 34031 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
| 9 | 2, 6, 8 | syl2anc 584 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
| 10 | 1, 9 | sselid 3981 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
| 11 | nfrab1 3457 | . . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} | |
| 12 | ssrab2 4080 | . . . . . 6 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴 | |
| 13 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴) |
| 14 | 0xr 11308 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
| 15 | pnfxr 11315 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
| 16 | 0lepnf 13175 | . . . . . . . 8 ⊢ 0 ≤ +∞ | |
| 17 | ubicc2 13505 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞)) | |
| 18 | 14, 15, 16, 17 | mp3an 1463 | . . . . . . 7 ⊢ +∞ ∈ (0[,]+∞) |
| 19 | 18 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ∈ (0[,]+∞)) |
| 20 | 0e0iccpnf 13499 | . . . . . . 7 ⊢ 0 ∈ (0[,]+∞) | |
| 21 | 20 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ∈ (0[,]+∞)) |
| 22 | 19, 21 | ifclda 4561 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ∈ (0[,]+∞)) |
| 23 | eldif 3961 | . . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) | |
| 24 | rabid 3458 | . . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = +∞)) | |
| 25 | 24 | simplbi2 500 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴 → (𝐵 = +∞ → 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) |
| 26 | 25 | con3dimp 408 | . . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞) |
| 27 | 23, 26 | sylbi 217 | . . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞) |
| 28 | 27 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ¬ 𝐵 = +∞) |
| 29 | 28 | iffalsed 4536 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → if(𝐵 = +∞, +∞, 0) = 0) |
| 30 | 3, 11, 7, 13, 2, 22, 29 | esumss 34073 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0)) |
| 31 | eqidd 2738 | . . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} = {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) | |
| 32 | 24 | simprbi 496 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → 𝐵 = +∞) |
| 33 | 32 | iftrued 4533 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → if(𝐵 = +∞, +∞, 0) = +∞) |
| 34 | 33 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → if(𝐵 = +∞, +∞, 0) = +∞) |
| 35 | 3, 31, 34 | esumeq12dvaf 34032 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞) |
| 36 | 2, 13 | ssexd 5324 | . . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V) |
| 37 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑘+∞ | |
| 38 | 11, 37 | esumcst 34064 | . . . . . 6 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ +∞ ∈ (0[,]+∞)) → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞)) |
| 39 | 36, 18, 38 | sylancl 586 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞)) |
| 40 | hashxrcl 14396 | . . . . . . 7 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V → (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ*) | |
| 41 | 36, 40 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ*) |
| 42 | esumpinfval.3 | . . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞) | |
| 43 | rabn0 4389 | . . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = +∞) | |
| 44 | 42, 43 | sylibr 234 | . . . . . . 7 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) |
| 45 | hashgt0 14427 | . . . . . . 7 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) | |
| 46 | 36, 44, 45 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) |
| 47 | xmulpnf1 13316 | . . . . . 6 ⊢ (((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ* ∧ 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞) = +∞) | |
| 48 | 41, 46, 47 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞) = +∞) |
| 49 | 35, 39, 48 | 3eqtrd 2781 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = +∞) |
| 50 | 30, 49 | eqtr3d 2779 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) = +∞) |
| 51 | breq1 5146 | . . . . 5 ⊢ (+∞ = if(𝐵 = +∞, +∞, 0) → (+∞ ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵)) | |
| 52 | breq1 5146 | . . . . 5 ⊢ (0 = if(𝐵 = +∞, +∞, 0) → (0 ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵)) | |
| 53 | pnfge 13172 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
| 54 | 15, 53 | ax-mp 5 | . . . . . . 7 ⊢ +∞ ≤ +∞ |
| 55 | breq2 5147 | . . . . . . 7 ⊢ (𝐵 = +∞ → (+∞ ≤ 𝐵 ↔ +∞ ≤ +∞)) | |
| 56 | 54, 55 | mpbiri 258 | . . . . . 6 ⊢ (𝐵 = +∞ → +∞ ≤ 𝐵) |
| 57 | 56 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ≤ 𝐵) |
| 58 | 4 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ (0[,]+∞)) |
| 59 | iccgelb 13443 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
| 60 | 14, 15, 59 | mp3an12 1453 | . . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 0 ≤ 𝐵) |
| 61 | 58, 60 | syl 17 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ≤ 𝐵) |
| 62 | 51, 52, 57, 61 | ifbothda 4564 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ≤ 𝐵) |
| 63 | 3, 7, 2, 22, 4, 62 | esumlef 34063 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) ≤ Σ*𝑘 ∈ 𝐴𝐵) |
| 64 | 50, 63 | eqbrtrrd 5167 | . 2 ⊢ (𝜑 → +∞ ≤ Σ*𝑘 ∈ 𝐴𝐵) |
| 65 | xgepnf 13207 | . . 3 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* → (+∞ ≤ Σ*𝑘 ∈ 𝐴𝐵 ↔ Σ*𝑘 ∈ 𝐴𝐵 = +∞)) | |
| 66 | 65 | biimpd 229 | . 2 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* → (+∞ ≤ Σ*𝑘 ∈ 𝐴𝐵 → Σ*𝑘 ∈ 𝐴𝐵 = +∞)) |
| 67 | 10, 64, 66 | sylc 65 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = +∞) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 ifcif 4525 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 ·e cxmu 13153 [,]cicc 13390 ♯chash 14369 Σ*cesum 34028 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-ordt 17546 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-ps 18611 df-tsr 18612 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-abv 20810 df-lmod 20860 df-scaf 20861 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tmd 24080 df-tgp 24081 df-tsms 24135 df-trg 24168 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 df-ii 24903 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-esum 34029 |
| This theorem is referenced by: hasheuni 34086 esumcvg 34087 esumcvgre 34092 voliune 34230 volfiniune 34231 |
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