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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0lempt | Structured version Visualization version GIF version |
Description: If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0lempt.xph | ⊢ Ⅎ𝑥𝜑 |
sge0lempt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sge0lempt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
sge0lempt.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) |
sge0lempt.le | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
sge0lempt | ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≤ (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0lempt.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | sge0lempt.xph | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | sge0lempt.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
4 | eqid 2725 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | 2, 3, 4 | fmptdf 7122 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
6 | sge0lempt.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) | |
7 | eqid 2725 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
8 | 2, 6, 7 | fmptdf 7122 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶(0[,]+∞)) |
9 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | |
10 | 2, 9 | nfan 1894 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐴) |
11 | nfcv 2892 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
12 | 11 | nfcsb1 3908 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 |
13 | nfcv 2892 | . . . . . 6 ⊢ Ⅎ𝑥 ≤ | |
14 | 11 | nfcsb1 3908 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 |
15 | 12, 13, 14 | nfbr 5190 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ≤ ⦋𝑦 / 𝑥⦌𝐶 |
16 | 10, 15 | nfim 1891 | . . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ≤ ⦋𝑦 / 𝑥⦌𝐶) |
17 | eleq1w 2808 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | 17 | anbi2d 628 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴))) |
19 | csbeq1a 3898 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
20 | csbeq1a 3898 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
21 | 19, 20 | breq12d 5156 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ≤ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ≤ ⦋𝑦 / 𝑥⦌𝐶)) |
22 | 18, 21 | imbi12d 343 | . . . 4 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ≤ ⦋𝑦 / 𝑥⦌𝐶))) |
23 | sge0lempt.le | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶) | |
24 | 16, 22, 23 | chvarfv 2228 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ≤ ⦋𝑦 / 𝑥⦌𝐶) |
25 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
26 | 12 | nfel1 2909 | . . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ (0[,]+∞) |
27 | 10, 26 | nfim 1891 | . . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ (0[,]+∞)) |
28 | 19 | eleq1d 2810 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ (0[,]+∞) ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ (0[,]+∞))) |
29 | 18, 28 | imbi12d 343 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ (0[,]+∞)))) |
30 | 27, 29, 3 | chvarfv 2228 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ (0[,]+∞)) |
31 | 11, 12, 19, 4 | fvmptf 7021 | . . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ ⦋𝑦 / 𝑥⦌𝐵 ∈ (0[,]+∞)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ⦋𝑦 / 𝑥⦌𝐵) |
32 | 25, 30, 31 | syl2anc 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ⦋𝑦 / 𝑥⦌𝐵) |
33 | nfcv 2892 | . . . . . . . 8 ⊢ Ⅎ𝑥(0[,]+∞) | |
34 | 14, 33 | nfel 2907 | . . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ (0[,]+∞) |
35 | 10, 34 | nfim 1891 | . . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (0[,]+∞)) |
36 | 20 | eleq1d 2810 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ (0[,]+∞) ↔ ⦋𝑦 / 𝑥⦌𝐶 ∈ (0[,]+∞))) |
37 | 18, 36 | imbi12d 343 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (0[,]+∞)) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (0[,]+∞)))) |
38 | 35, 37, 6 | chvarfv 2228 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (0[,]+∞)) |
39 | 11, 14, 20, 7 | fvmptf 7021 | . . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ ⦋𝑦 / 𝑥⦌𝐶 ∈ (0[,]+∞)) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = ⦋𝑦 / 𝑥⦌𝐶) |
40 | 25, 38, 39 | syl2anc 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = ⦋𝑦 / 𝑥⦌𝐶) |
41 | 32, 40 | breq12d 5156 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) ↔ ⦋𝑦 / 𝑥⦌𝐵 ≤ ⦋𝑦 / 𝑥⦌𝐶)) |
42 | 24, 41 | mpbird 256 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)) |
43 | 1, 5, 8, 42 | sge0le 45858 | 1 ⊢ (𝜑 → (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≤ (Σ^‘(𝑥 ∈ 𝐴 ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ⦋csb 3884 class class class wbr 5143 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7416 0cc0 11138 +∞cpnf 11275 ≤ cle 11279 [,]cicc 13359 Σ^csumge0 45813 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 df-sumge0 45814 |
This theorem is referenced by: sge0iunmptlemre 45866 sge0xadd 45886 meaiunlelem 45919 hoicvrrex 46007 ovnsubaddlem1 46021 sge0hsphoire 46040 hoidmv1lelem1 46042 hoidmv1lelem2 46043 hoidmv1lelem3 46044 hoidmvlelem1 46046 hoidmvlelem2 46047 hoidmvlelem4 46049 hspmbllem2 46078 ovolval5lem1 46103 |
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