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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0iun | Structured version Visualization version GIF version |
Description: Sum of nonnegative extended reals over a disjoint indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0iun.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0iun.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
sge0iun.x | ⊢ 𝑋 = ∪ 𝑥 ∈ 𝐴 𝐵 |
sge0iun.f | ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) |
sge0iun.dj | ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) |
Ref | Expression |
---|---|
sge0iun | ⊢ (𝜑 → (Σ^‘𝐹) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝐹 ↾ 𝐵))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0iun.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sge0iun.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) | |
3 | sge0iun.dj | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) | |
4 | sge0iun.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞)) | |
5 | 4 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝑋⟶(0[,]+∞)) |
6 | 5 | 3adant3 1129 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐹:𝑋⟶(0[,]+∞)) |
7 | ssiun2 5051 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
8 | 7 | adantl 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
9 | sge0iun.x | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝑥 ∈ 𝐴 𝐵 | |
10 | 9 | eqcomi 2734 | . . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = 𝑋 |
11 | 8, 10 | sseqtrdi 4027 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝑋) |
12 | 11 | 3adant3 1129 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐵 ⊆ 𝑋) |
13 | simp3 1135 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) | |
14 | 12, 13 | sseldd 3977 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝑋) |
15 | 6, 14 | ffvelcdmd 7094 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ (0[,]+∞)) |
16 | 1, 2, 3, 15 | sge0iunmpt 45944 | . 2 ⊢ (𝜑 → (Σ^‘(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (𝐹‘𝑦))) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦)))))) |
17 | 9 | feq2i 6715 | . . . . . 6 ⊢ (𝐹:𝑋⟶(0[,]+∞) ↔ 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶(0[,]+∞)) |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐹:𝑋⟶(0[,]+∞) ↔ 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶(0[,]+∞))) |
19 | 4, 18 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶(0[,]+∞)) |
20 | 19 | feqmptd 6966 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (𝐹‘𝑦))) |
21 | 20 | fveq2d 6900 | . 2 ⊢ (𝜑 → (Σ^‘𝐹) = (Σ^‘(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (𝐹‘𝑦)))) |
22 | 5, 11 | fssresd 6764 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞)) |
23 | 22 | feqmptd 6966 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ((𝐹 ↾ 𝐵)‘𝑦))) |
24 | fvres 6915 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑦) = (𝐹‘𝑦)) | |
25 | 24 | mpteq2ia 5252 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐵 ↦ ((𝐹 ↾ 𝐵)‘𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦)) |
26 | 25 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↦ ((𝐹 ↾ 𝐵)‘𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦))) |
27 | 23, 26 | eqtrd 2765 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦))) |
28 | 27 | fveq2d 6900 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (Σ^‘(𝐹 ↾ 𝐵)) = (Σ^‘(𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦)))) |
29 | 28 | mpteq2dva 5249 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (Σ^‘(𝐹 ↾ 𝐵))) = (𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦))))) |
30 | 29 | fveq2d 6900 | . 2 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝐹 ↾ 𝐵)))) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝑦 ∈ 𝐵 ↦ (𝐹‘𝑦)))))) |
31 | 16, 21, 30 | 3eqtr4d 2775 | 1 ⊢ (𝜑 → (Σ^‘𝐹) = (Σ^‘(𝑥 ∈ 𝐴 ↦ (Σ^‘(𝐹 ↾ 𝐵))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ∪ ciun 4997 Disj wdisj 5114 ↦ cmpt 5232 ↾ cres 5680 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 0cc0 11140 +∞cpnf 11277 [,]cicc 13362 Σ^csumge0 45888 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-ac2 10488 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-oi 9535 df-card 9964 df-acn 9967 df-ac 10141 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-xadd 13128 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-sum 15669 df-sumge0 45889 |
This theorem is referenced by: psmeasurelem 45996 |
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