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 1911 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | fveq2 6664 | . . . 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 6683 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → ((𝐺 ↾ 𝑋)‘𝑥) = (𝐺‘𝑥)) | |
7 | 6 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐺 ↾ 𝑋)‘𝑥) = (𝐺‘𝑥)) |
8 | sge0resrnlem.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞)) | |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝐹:𝐵⟶(0[,]+∞)) |
10 | sge0resrnlem.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
11 | 10 | frnd 6515 | . . . . . . 7 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐵) |
12 | 11 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → ran 𝐺 ⊆ 𝐵) |
13 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ∈ ran 𝐺) | |
14 | 12, 13 | sseldd 3967 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ∈ 𝐵) |
15 | 9, 14 | ffvelrnd 6846 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → (𝐹‘𝑦) ∈ (0[,]+∞)) |
16 | 1, 2, 3, 4, 5, 7, 15 | sge0f1o 42658 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑦 ∈ ran 𝐺 ↦ (𝐹‘𝑦))) = (Σ^‘(𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥))))) |
17 | 8, 11 | feqresmpt 6728 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺) = (𝑦 ∈ ran 𝐺 ↦ (𝐹‘𝑦))) |
18 | 17 | fveq2d 6668 | . . 3 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) = (Σ^‘(𝑦 ∈ ran 𝐺 ↦ (𝐹‘𝑦)))) |
19 | fcompt 6889 | . . . . . . 7 ⊢ ((𝐹:𝐵⟶(0[,]+∞) ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥)))) | |
20 | 8, 10, 19 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥)))) |
21 | 20 | reseq1d 5846 | . . . . 5 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ↾ 𝑋) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥))) ↾ 𝑋)) |
22 | 4 | elpwid 4552 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝐴) |
23 | 22 | resmptd 5902 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥))) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥)))) |
24 | 21, 23 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥)))) |
25 | 24 | fveq2d 6668 | . . 3 ⊢ (𝜑 → (Σ^‘((𝐹 ∘ 𝐺) ↾ 𝑋)) = (Σ^‘(𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥))))) |
26 | 16, 18, 25 | 3eqtr4d 2866 | . 2 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) = (Σ^‘((𝐹 ∘ 𝐺) ↾ 𝑋))) |
27 | sge0resrnlem.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
28 | fco 6525 | . . . 4 ⊢ ((𝐹:𝐵⟶(0[,]+∞) ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶(0[,]+∞)) | |
29 | 8, 10, 28 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶(0[,]+∞)) |
30 | 27, 29 | sge0less 42668 | . 2 ⊢ (𝜑 → (Σ^‘((𝐹 ∘ 𝐺) ↾ 𝑋)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
31 | 26, 30 | eqbrtrd 5080 | 1 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 𝒫 cpw 4538 class class class wbr 5058 ↦ cmpt 5138 ran crn 5550 ↾ cres 5551 ∘ ccom 5553 ⟶wf 6345 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 0cc0 10531 +∞cpnf 10666 ≤ cle 10670 [,]cicc 12735 Σ^csumge0 42638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-sumge0 42639 |
This theorem is referenced by: sge0resrn 42680 |
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