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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 1912 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | fveq2 6907 | . . . 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 6926 | . . . . 5 ⊢ (𝑥 ∈ 𝑋 → ((𝐺 ↾ 𝑋)‘𝑥) = (𝐺‘𝑥)) | |
7 | 6 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐺 ↾ 𝑋)‘𝑥) = (𝐺‘𝑥)) |
8 | sge0resrnlem.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞)) | |
9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝐹:𝐵⟶(0[,]+∞)) |
10 | sge0resrnlem.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
11 | 10 | frnd 6745 | . . . . . . 7 ⊢ (𝜑 → ran 𝐺 ⊆ 𝐵) |
12 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → ran 𝐺 ⊆ 𝐵) |
13 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ∈ ran 𝐺) | |
14 | 12, 13 | sseldd 3996 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ∈ 𝐵) |
15 | 9, 14 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → (𝐹‘𝑦) ∈ (0[,]+∞)) |
16 | 1, 2, 3, 4, 5, 7, 15 | sge0f1o 46338 | . . 3 ⊢ (𝜑 → (Σ^‘(𝑦 ∈ ran 𝐺 ↦ (𝐹‘𝑦))) = (Σ^‘(𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥))))) |
17 | 8, 11 | feqresmpt 6978 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ ran 𝐺) = (𝑦 ∈ ran 𝐺 ↦ (𝐹‘𝑦))) |
18 | 17 | fveq2d 6911 | . . 3 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) = (Σ^‘(𝑦 ∈ ran 𝐺 ↦ (𝐹‘𝑦)))) |
19 | fcompt 7153 | . . . . . . 7 ⊢ ((𝐹:𝐵⟶(0[,]+∞) ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥)))) | |
20 | 8, 10, 19 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥)))) |
21 | 20 | reseq1d 5999 | . . . . 5 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ↾ 𝑋) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥))) ↾ 𝑋)) |
22 | 4 | elpwid 4614 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝐴) |
23 | 22 | resmptd 6060 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘(𝐺‘𝑥))) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥)))) |
24 | 21, 23 | eqtrd 2775 | . . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐺) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥)))) |
25 | 24 | fveq2d 6911 | . . 3 ⊢ (𝜑 → (Σ^‘((𝐹 ∘ 𝐺) ↾ 𝑋)) = (Σ^‘(𝑥 ∈ 𝑋 ↦ (𝐹‘(𝐺‘𝑥))))) |
26 | 16, 18, 25 | 3eqtr4d 2785 | . 2 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) = (Σ^‘((𝐹 ∘ 𝐺) ↾ 𝑋))) |
27 | sge0resrnlem.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
28 | fco 6761 | . . . 4 ⊢ ((𝐹:𝐵⟶(0[,]+∞) ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶(0[,]+∞)) | |
29 | 8, 10, 28 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶(0[,]+∞)) |
30 | 27, 29 | sge0less 46348 | . 2 ⊢ (𝜑 → (Σ^‘((𝐹 ∘ 𝐺) ↾ 𝑋)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
31 | 26, 30 | eqbrtrd 5170 | 1 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 𝒫 cpw 4605 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ↾ cres 5691 ∘ ccom 5693 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 ≤ cle 11294 [,]cicc 13387 Σ^csumge0 46318 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-sumge0 46319 |
This theorem is referenced by: sge0resrn 46360 |
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