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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0resrnlem | Structured version Visualization version GIF version |
Description: The sum of nonnegative extended reals restricted to the range of a function is less than or equal to the sum of the composition of the two functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0resrnlem.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0resrnlem.f | ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞)) |
sge0resrnlem.g | ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
sge0resrnlem.x | ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝐴) |
sge0resrnlem.f1o | ⊢ (𝜑 → (𝐺 ↾ 𝑋):𝑋–1-1-onto→ran 𝐺) |
Ref | Expression |
---|---|
sge0resrnlem | ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | fveq2 6645 | . . . 4 ⊢ (𝑦 = (𝐺‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐺‘𝑥))) | |
4 | sge0resrnlem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝐴) | |
5 | sge0resrnlem.f1o | . . . 4 ⊢ (𝜑 → (𝐺 ↾ 𝑋):𝑋–1-1-onto→ran 𝐺) | |
6 | fvres 6664 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → ((𝐺 ↾ 𝑋)‘𝑥) = (𝐺‘𝑥)) | |
7 | 6 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐺 ↾ 𝑋)‘𝑥) = (𝐺‘𝑥)) |
8 | sge0resrnlem.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞)) | |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝐹:𝐵⟶(0[,]+∞)) |
10 | sge0resrnlem.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
11 | 10 | frnd 6494 | . . . . . . 7 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐵) |
12 | 11 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → ran 𝐺 ⊆ 𝐵) |
13 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ∈ ran 𝐺) | |
14 | 12, 13 | sseldd 3916 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ∈ 𝐵) |
15 | 9, 14 | ffvelrnd 6829 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → (𝐹‘𝑦) ∈ (0[,]+∞)) |
16 | 1, 2, 3, 4, 5, 7, 15 | sge0f1o 43021 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑦 ∈ ran 𝐺 ↦ (𝐹‘𝑦))) = (Σ^‘(𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥))))) |
17 | 8, 11 | feqresmpt 6709 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺) = (𝑦 ∈ ran 𝐺 ↦ (𝐹‘𝑦))) |
18 | 17 | fveq2d 6649 | . . 3 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) = (Σ^‘(𝑦 ∈ ran 𝐺 ↦ (𝐹‘𝑦)))) |
19 | fcompt 6872 | . . . . . . 7 ⊢ ((𝐹:𝐵⟶(0[,]+∞) ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥)))) | |
20 | 8, 10, 19 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥)))) |
21 | 20 | reseq1d 5817 | . . . . 5 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ↾ 𝑋) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥))) ↾ 𝑋)) |
22 | 4 | elpwid 4508 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝐴) |
23 | 22 | resmptd 5875 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥))) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥)))) |
24 | 21, 23 | eqtrd 2833 | . . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥)))) |
25 | 24 | fveq2d 6649 | . . 3 ⊢ (𝜑 → (Σ^‘((𝐹 ∘ 𝐺) ↾ 𝑋)) = (Σ^‘(𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥))))) |
26 | 16, 18, 25 | 3eqtr4d 2843 | . 2 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) = (Σ^‘((𝐹 ∘ 𝐺) ↾ 𝑋))) |
27 | sge0resrnlem.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
28 | fco 6505 | . . . 4 ⊢ ((𝐹:𝐵⟶(0[,]+∞) ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶(0[,]+∞)) | |
29 | 8, 10, 28 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶(0[,]+∞)) |
30 | 27, 29 | sge0less 43031 | . 2 ⊢ (𝜑 → (Σ^‘((𝐹 ∘ 𝐺) ↾ 𝑋)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
31 | 26, 30 | eqbrtrd 5052 | 1 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4497 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ↾ cres 5521 ∘ ccom 5523 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 0cc0 10526 +∞cpnf 10661 ≤ cle 10665 [,]cicc 12729 Σ^csumge0 43001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-sumge0 43002 |
This theorem is referenced by: sge0resrn 43043 |
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