Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0ss | Structured version Visualization version GIF version |
Description: Change the index set to a subset in a sum of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0ss.kph | ⊢ Ⅎ𝑘𝜑 |
sge0ss.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
sge0ss.a | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sge0ss.c | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) |
sge0ss.c0 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) |
Ref | Expression |
---|---|
sge0ss | ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) = (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0ss.kph | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
2 | sge0ss.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
3 | sge0ss.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | ssexg 5225 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V) | |
5 | 2, 3, 4 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
6 | 3 | difexd 5231 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ V) |
7 | disjdif 4395 | . . . . 5 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅) |
9 | sge0ss.c | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
10 | sge0ss.c0 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) | |
11 | 0e0iccpnf 13060 | . . . . . 6 ⊢ 0 ∈ (0[,]+∞) | |
12 | 11 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 0 ∈ (0[,]+∞)) |
13 | 10, 12 | eqeltrd 2839 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ (0[,]+∞)) |
14 | 1, 5, 6, 8, 9, 13 | sge0splitmpt 43639 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) ↦ 𝐶)) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 (Σ^‘(𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)))) |
15 | 14 | eqcomd 2744 | . 2 ⊢ (𝜑 → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 (Σ^‘(𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶))) = (Σ^‘(𝑘 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) ↦ 𝐶))) |
16 | 1, 10 | mpteq2da 5158 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 0)) |
17 | 16 | fveq2d 6730 | . . . . 5 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)) = (Σ^‘(𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 0))) |
18 | 1, 6 | sge0z 43603 | . . . . 5 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 0)) = 0) |
19 | 17, 18 | eqtrd 2778 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)) = 0) |
20 | 19 | oveq2d 7238 | . . 3 ⊢ (𝜑 → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 (Σ^‘(𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶))) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 0)) |
21 | eqid 2738 | . . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶) | |
22 | 1, 9, 21 | fmptdf 6943 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶(0[,]+∞)) |
23 | 5, 22 | sge0xrcl 43613 | . . . 4 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) ∈ ℝ*) |
24 | xaddid1 12844 | . . . 4 ⊢ ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) ∈ ℝ* → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 0) = (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶))) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ (𝜑 → ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 0) = (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶))) |
26 | eqidd 2739 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) = (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶))) | |
27 | 20, 25, 26 | 3eqtrrd 2783 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) = ((Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) +𝑒 (Σ^‘(𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)))) |
28 | undif 4405 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) | |
29 | 2, 28 | sylib 221 | . . . . 5 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵) |
30 | 29 | eqcomd 2744 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝐴 ∪ (𝐵 ∖ 𝐴))) |
31 | 30 | mpteq1d 5153 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) ↦ 𝐶)) |
32 | 31 | fveq2d 6730 | . 2 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶)) = (Σ^‘(𝑘 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) ↦ 𝐶))) |
33 | 15, 27, 32 | 3eqtr4d 2788 | 1 ⊢ (𝜑 → (Σ^‘(𝑘 ∈ 𝐴 ↦ 𝐶)) = (Σ^‘(𝑘 ∈ 𝐵 ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2111 Vcvv 3415 ∖ cdif 3872 ∪ cun 3873 ∩ cin 3874 ⊆ wss 3875 ∅c0 4246 ↦ cmpt 5144 ‘cfv 6389 (class class class)co 7222 0cc0 10742 +∞cpnf 10877 ℝ*cxr 10879 +𝑒 cxad 12715 [,]cicc 12951 Σ^csumge0 43590 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 ax-inf2 9269 ax-cnex 10798 ax-resscn 10799 ax-1cn 10800 ax-icn 10801 ax-addcl 10802 ax-addrcl 10803 ax-mulcl 10804 ax-mulrcl 10805 ax-mulcom 10806 ax-addass 10807 ax-mulass 10808 ax-distr 10809 ax-i2m1 10810 ax-1ne0 10811 ax-1rid 10812 ax-rnegex 10813 ax-rrecex 10814 ax-cnre 10815 ax-pre-lttri 10816 ax-pre-lttrn 10817 ax-pre-ltadd 10818 ax-pre-mulgt0 10819 ax-pre-sup 10820 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-int 4869 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-se 5519 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-isom 6398 df-riota 7179 df-ov 7225 df-oprab 7226 df-mpo 7227 df-om 7654 df-1st 7770 df-2nd 7771 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-er 8400 df-en 8636 df-dom 8637 df-sdom 8638 df-fin 8639 df-sup 9071 df-oi 9139 df-card 9568 df-pnf 10882 df-mnf 10883 df-xr 10884 df-ltxr 10885 df-le 10886 df-sub 11077 df-neg 11078 df-div 11503 df-nn 11844 df-2 11906 df-3 11907 df-n0 12104 df-z 12190 df-uz 12452 df-rp 12600 df-xadd 12718 df-ico 12954 df-icc 12955 df-fz 13109 df-fzo 13252 df-seq 13588 df-exp 13649 df-hash 13910 df-cj 14675 df-re 14676 df-im 14677 df-sqrt 14811 df-abs 14812 df-clim 15062 df-sum 15263 df-sumge0 43591 |
This theorem is referenced by: sge0fodjrnlem 43644 meadjiunlem 43693 ovnhoilem1 43829 ovnsubadd2lem 43873 |
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