sitmf - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > sitmf		Structured version Visualization version GIF version

Theorem sitmf 32327

Description: The integral metric as a function. (Contributed by Thierry Arnoux, 13-Mar-2018.)

**Hypotheses**
Ref	Expression
sitmf.0	⊢ (𝜑 → 𝑊 ∈ Mnd)
sitmf.1	⊢ (𝜑 → 𝑊 ∈ ∞MetSp)
sitmf.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)

**Assertion**
Ref	Expression
sitmf	⊢ (𝜑 → (𝑊sitm𝑀):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞))

Proof of Theorem sitmf Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . . 6 ⊢ (dist‘𝑊) = (dist‘𝑊)
2		sitmf.1	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ∞MetSp)
3	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑊 ∈ ∞MetSp)
4		sitmf.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
5	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑀 ∈ ∪ ran measures)
6		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑓 ∈ dom (𝑊sitg𝑀))
7		simprr 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑔 ∈ dom (𝑊sitg𝑀))
8	1, 3, 5, 6, 7	sitmfval 32325	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → (𝑓(𝑊sitm𝑀)𝑔) = (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔)))
9		sitmf.0	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Mnd)
10	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑊 ∈ Mnd)
11	10, 3, 5, 6, 7	sitmcl 32326	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → (𝑓(𝑊sitm𝑀)𝑔) ∈ (0[,]+∞))
12	8, 11	eqeltrrd 2840	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔)) ∈ (0[,]+∞))
13	12	ralrimivva 3115	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)∀𝑔 ∈ dom (𝑊sitg𝑀)(((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔)) ∈ (0[,]+∞))
14		eqid 2738	. . . 4 ⊢ (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ^_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔))) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ^_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔)))
15	14	fmpo 7897	. . 3 ⊢ (∀𝑓 ∈ dom (𝑊sitg𝑀)∀𝑔 ∈ dom (𝑊sitg𝑀)(((ℝ^_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔)) ∈ (0[,]+∞) ↔ (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ^_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔))):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞))
16	13, 15	sylib 217	. 2 ⊢ (𝜑 → (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔))):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞))
17	1, 2, 4	sitmval 32324	. . 3 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔))))
18	17	feq1d 6577	. 2 ⊢ (𝜑 → ((𝑊sitm𝑀):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞) ↔ (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔))):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞)))
19	16, 18	mpbird 256	1 ⊢ (𝜑 → (𝑊sitm𝑀):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 ∪ cuni 4839 × cxp 5582 dom cdm 5584 ran crn 5585 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 ∈ cmpo 7269 ∘_f cof 7521 0cc0 10881 +∞cpnf 11016 [,]cicc 13092 ↾_s cress 16951 distcds 16981 ℝ^*_𝑠cxrs 17221 Mndcmnd 18395 ∞MetSpcxms 23480 measurescmeas 32171 sitmcsitm 32303 sitgcsitg 32304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-ac2 10229 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 ax-addf 10960 ax-mulf 10961 ax-xrssca 31290 ax-xrsvsca 31291

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5039 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-tpos 8029 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-fi 9157 df-sup 9188 df-inf 9189 df-oi 9256 df-dju 9669 df-card 9707 df-acn 9710 df-ac 9882 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-ioo 13093 df-ioc 13094 df-ico 13095 df-icc 13096 df-fz 13250 df-fzo 13393 df-fl 13522 df-mod 13600 df-seq 13732 df-exp 13793 df-fac 13998 df-bc 14027 df-hash 14055 df-shft 14788 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-limsup 15190 df-clim 15207 df-rlim 15208 df-sum 15408 df-ef 15787 df-sin 15789 df-cos 15790 df-pi 15792 df-dvds 15974 df-gcd 16212 df-numer 16449 df-denom 16450 df-gz 16641 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-hom 16996 df-cco 16997 df-rest 17143 df-topn 17144 df-0g 17162 df-gsum 17163 df-topgen 17164 df-pt 17165 df-prds 17168 df-ordt 17222 df-xrs 17223 df-qtop 17228 df-imas 17229 df-xps 17231 df-mre 17305 df-mrc 17306 df-acs 17308 df-proset 18023 df-poset 18041 df-plt 18058 df-toset 18145 df-ps 18294 df-tsr 18295 df-plusf 18335 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-mhm 18440 df-submnd 18441 df-grp 18590 df-minusg 18591 df-sbg 18592 df-mulg 18711 df-subg 18762 df-ghm 18842 df-cntz 18933 df-od 19146 df-cmn 19398 df-abl 19399 df-mgp 19731 df-ur 19748 df-ring 19795 df-cring 19796 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-dvr 19935 df-rnghom 19969 df-drng 20003 df-field 20004 df-subrg 20032 df-abv 20087 df-lmod 20135 df-scaf 20136 df-sra 20444 df-rgmod 20445 df-nzr 20539 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-fbas 20604 df-fg 20605 df-metu 20606 df-cnfld 20608 df-zring 20681 df-zrh 20715 df-zlm 20716 df-chr 20717 df-refld 20820 df-top 22053 df-topon 22070 df-topsp 22092 df-bases 22106 df-cld 22180 df-ntr 22181 df-cls 22182 df-nei 22259 df-lp 22297 df-perf 22298 df-cn 22388 df-cnp 22389 df-t1 22475 df-haus 22476 df-reg 22477 df-cmp 22548 df-tx 22723 df-hmeo 22916 df-fil 23007 df-fm 23099 df-flim 23100 df-flf 23101 df-fcls 23102 df-cnext 23221 df-tmd 23233 df-tgp 23234 df-tsms 23288 df-trg 23321 df-ust 23362 df-utop 23393 df-uss 23418 df-usp 23419 df-ucn 23438 df-cfilu 23449 df-cusp 23460 df-xms 23483 df-ms 23484 df-tms 23485 df-nm 23748 df-ngp 23749 df-nrg 23751 df-nlm 23752 df-ii 24050 df-cncf 24051 df-cfil 24429 df-cmet 24431 df-cms 24509 df-limc 25040 df-dv 25041 df-log 25722 df-omnd 31333 df-ogrp 31334 df-orng 31504 df-ofld 31505 df-qqh 31931 df-rrh 31953 df-rrext 31957 df-esum 32004 df-siga 32085 df-sigagen 32115 df-meas 32172 df-mbfm 32226 df-sitg 32305 df-sitm 32306

This theorem is referenced by: (None)

W3C validator