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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0resrn | 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 (well-order hypothesis allows to avoid using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0resrn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0resrn.f | ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞)) |
sge0resrn.g | ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
sge0resrn.r | ⊢ (𝜑 → 𝑅 We 𝐴) |
Ref | Expression |
---|---|
sge0resrn | ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0resrn.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
2 | 1 | ffnd 6666 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) |
3 | sge0resrn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | sge0resrn.r | . . 3 ⊢ (𝜑 → 𝑅 We 𝐴) | |
5 | 2, 3, 4 | wessf1orn 43303 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝒫 𝐴(𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺) |
6 | 3 | 3ad2ant1 1133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ (𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺) → 𝐴 ∈ 𝑉) |
7 | sge0resrn.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶(0[,]+∞)) | |
8 | 7 | 3ad2ant1 1133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ (𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺) → 𝐹:𝐵⟶(0[,]+∞)) |
9 | 1 | 3ad2ant1 1133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ (𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺) → 𝐺:𝐴⟶𝐵) |
10 | simp2 1137 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ (𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺) → 𝑥 ∈ 𝒫 𝐴) | |
11 | simp3 1138 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ (𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺) → (𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺) | |
12 | 6, 8, 9, 10, 11 | sge0resrnlem 44539 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 𝐴 ∧ (𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺) → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
13 | 12 | 3exp 1119 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → ((𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺))))) |
14 | 13 | rexlimdv 3148 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝒫 𝐴(𝐺 ↾ 𝑥):𝑥–1-1-onto→ran 𝐺 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺)))) |
15 | 5, 14 | mpd 15 | 1 ⊢ (𝜑 → (Σ^‘(𝐹 ↾ ran 𝐺)) ≤ (Σ^‘(𝐹 ∘ 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 ∈ wcel 2106 ∃wrex 3071 𝒫 cpw 4558 class class class wbr 5103 We wwe 5585 ran crn 5632 ↾ cres 5633 ∘ ccom 5635 ⟶wf 6489 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7351 0cc0 11009 +∞cpnf 11144 ≤ cle 11148 [,]cicc 13221 Σ^csumge0 44498 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-seq 13861 df-exp 13922 df-hash 14185 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-clim 15324 df-sum 15525 df-sumge0 44499 |
This theorem is referenced by: omeiunle 44653 |
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