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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 | β’ (π β π΄ β π) |
sge0resrnlem.f | β’ (π β πΉ:π΅βΆ(0[,]+β)) |
sge0resrnlem.g | β’ (π β πΊ:π΄βΆπ΅) |
sge0resrnlem.x | β’ (π β π β π« π΄) |
sge0resrnlem.f1o | β’ (π β (πΊ βΎ π):πβ1-1-ontoβran πΊ) |
Ref | Expression |
---|---|
sge0resrnlem | β’ (π β (Ξ£^β(πΉ βΎ ran πΊ)) β€ (Ξ£^β(πΉ β πΊ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1910 | . . . 4 β’ β²π¦π | |
2 | nfv 1910 | . . . 4 β’ β²π₯π | |
3 | fveq2 6891 | . . . 4 β’ (π¦ = (πΊβπ₯) β (πΉβπ¦) = (πΉβ(πΊβπ₯))) | |
4 | sge0resrnlem.x | . . . 4 β’ (π β π β π« π΄) | |
5 | sge0resrnlem.f1o | . . . 4 β’ (π β (πΊ βΎ π):πβ1-1-ontoβran πΊ) | |
6 | fvres 6910 | . . . . 5 β’ (π₯ β π β ((πΊ βΎ π)βπ₯) = (πΊβπ₯)) | |
7 | 6 | adantl 481 | . . . 4 β’ ((π β§ π₯ β π) β ((πΊ βΎ π)βπ₯) = (πΊβπ₯)) |
8 | sge0resrnlem.f | . . . . . 6 β’ (π β πΉ:π΅βΆ(0[,]+β)) | |
9 | 8 | adantr 480 | . . . . 5 β’ ((π β§ π¦ β ran πΊ) β πΉ:π΅βΆ(0[,]+β)) |
10 | sge0resrnlem.g | . . . . . . . 8 β’ (π β πΊ:π΄βΆπ΅) | |
11 | 10 | frnd 6724 | . . . . . . 7 β’ (π β ran πΊ β π΅) |
12 | 11 | adantr 480 | . . . . . 6 β’ ((π β§ π¦ β ran πΊ) β ran πΊ β π΅) |
13 | simpr 484 | . . . . . 6 β’ ((π β§ π¦ β ran πΊ) β π¦ β ran πΊ) | |
14 | 12, 13 | sseldd 3979 | . . . . 5 β’ ((π β§ π¦ β ran πΊ) β π¦ β π΅) |
15 | 9, 14 | ffvelcdmd 7089 | . . . 4 β’ ((π β§ π¦ β ran πΊ) β (πΉβπ¦) β (0[,]+β)) |
16 | 1, 2, 3, 4, 5, 7, 15 | sge0f1o 45683 | . . 3 β’ (π β (Ξ£^β(π¦ β ran πΊ β¦ (πΉβπ¦))) = (Ξ£^β(π₯ β π β¦ (πΉβ(πΊβπ₯))))) |
17 | 8, 11 | feqresmpt 6962 | . . . 4 β’ (π β (πΉ βΎ ran πΊ) = (π¦ β ran πΊ β¦ (πΉβπ¦))) |
18 | 17 | fveq2d 6895 | . . 3 β’ (π β (Ξ£^β(πΉ βΎ ran πΊ)) = (Ξ£^β(π¦ β ran πΊ β¦ (πΉβπ¦)))) |
19 | fcompt 7136 | . . . . . . 7 β’ ((πΉ:π΅βΆ(0[,]+β) β§ πΊ:π΄βΆπ΅) β (πΉ β πΊ) = (π₯ β π΄ β¦ (πΉβ(πΊβπ₯)))) | |
20 | 8, 10, 19 | syl2anc 583 | . . . . . 6 β’ (π β (πΉ β πΊ) = (π₯ β π΄ β¦ (πΉβ(πΊβπ₯)))) |
21 | 20 | reseq1d 5978 | . . . . 5 β’ (π β ((πΉ β πΊ) βΎ π) = ((π₯ β π΄ β¦ (πΉβ(πΊβπ₯))) βΎ π)) |
22 | 4 | elpwid 4607 | . . . . . 6 β’ (π β π β π΄) |
23 | 22 | resmptd 6038 | . . . . 5 β’ (π β ((π₯ β π΄ β¦ (πΉβ(πΊβπ₯))) βΎ π) = (π₯ β π β¦ (πΉβ(πΊβπ₯)))) |
24 | 21, 23 | eqtrd 2767 | . . . 4 β’ (π β ((πΉ β πΊ) βΎ π) = (π₯ β π β¦ (πΉβ(πΊβπ₯)))) |
25 | 24 | fveq2d 6895 | . . 3 β’ (π β (Ξ£^β((πΉ β πΊ) βΎ π)) = (Ξ£^β(π₯ β π β¦ (πΉβ(πΊβπ₯))))) |
26 | 16, 18, 25 | 3eqtr4d 2777 | . 2 β’ (π β (Ξ£^β(πΉ βΎ ran πΊ)) = (Ξ£^β((πΉ β πΊ) βΎ π))) |
27 | sge0resrnlem.a | . . 3 β’ (π β π΄ β π) | |
28 | fco 6741 | . . . 4 β’ ((πΉ:π΅βΆ(0[,]+β) β§ πΊ:π΄βΆπ΅) β (πΉ β πΊ):π΄βΆ(0[,]+β)) | |
29 | 8, 10, 28 | syl2anc 583 | . . 3 β’ (π β (πΉ β πΊ):π΄βΆ(0[,]+β)) |
30 | 27, 29 | sge0less 45693 | . 2 β’ (π β (Ξ£^β((πΉ β πΊ) βΎ π)) β€ (Ξ£^β(πΉ β πΊ))) |
31 | 26, 30 | eqbrtrd 5164 | 1 β’ (π β (Ξ£^β(πΉ βΎ ran πΊ)) β€ (Ξ£^β(πΉ β πΊ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3944 π« cpw 4598 class class class wbr 5142 β¦ cmpt 5225 ran crn 5673 βΎ cres 5674 β ccom 5676 βΆwf 6538 β1-1-ontoβwf1o 6541 βcfv 6542 (class class class)co 7414 0cc0 11124 +βcpnf 11261 β€ cle 11265 [,]cicc 13345 Ξ£^csumge0 45663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-sum 15651 df-sumge0 45664 |
This theorem is referenced by: sge0resrn 45705 |
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