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 32219

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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑊 ∈ ∞MetSp)
4		sitmf.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑀 ∈ ∪ ran measures)
6		simprl 767	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑓 ∈ dom (𝑊sitg𝑀))
7		simprr 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑔 ∈ dom (𝑊sitg𝑀))
8	1, 3, 5, 6, 7	sitmfval 32217	. . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑊 ∈ Mnd)
11	10, 3, 5, 6, 7	sitmcl 32218	. . . . 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 3114	. . 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 7881	. . 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 32216	. . 3 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔))))
18	17	feq1d 6569	. 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 395 ∈ wcel 2108 ∀wral 3063 ∪ cuni 4836 × cxp 5578 dom cdm 5580 ran crn 5581 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 ∘_f cof 7509 0cc0 10802 +∞cpnf 10937 [,]cicc 13011 ↾_s cress 16867 distcds 16897 ℝ^*_𝑠cxrs 17128 Mndcmnd 18300 ∞MetSpcxms 23378 measurescmeas 32063 sitmcsitm 32195 sitgcsitg 32196

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-ac2 10150 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882 ax-xrssca 31184 ax-xrsvsca 31185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-ac 9803 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-dvds 15892 df-gcd 16130 df-numer 16367 df-denom 16368 df-gz 16559 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-ordt 17129 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-proset 17928 df-poset 17946 df-plt 17963 df-toset 18050 df-ps 18199 df-tsr 18200 df-plusf 18240 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-od 19051 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-field 19909 df-subrg 19937 df-abv 19992 df-lmod 20040 df-scaf 20041 df-sra 20349 df-rgmod 20350 df-nzr 20442 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-metu 20509 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-zlm 20618 df-chr 20619 df-refld 20722 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-t1 22373 df-haus 22374 df-reg 22375 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-fcls 23000 df-cnext 23119 df-tmd 23131 df-tgp 23132 df-tsms 23186 df-trg 23219 df-ust 23260 df-utop 23291 df-uss 23316 df-usp 23317 df-ucn 23336 df-cfilu 23347 df-cusp 23358 df-xms 23381 df-ms 23382 df-tms 23383 df-nm 23644 df-ngp 23645 df-nrg 23647 df-nlm 23648 df-ii 23946 df-cncf 23947 df-cfil 24324 df-cmet 24326 df-cms 24404 df-limc 24935 df-dv 24936 df-log 25617 df-omnd 31227 df-ogrp 31228 df-orng 31398 df-ofld 31399 df-qqh 31823 df-rrh 31845 df-rrext 31849 df-esum 31896 df-siga 31977 df-sigagen 32007 df-meas 32064 df-mbfm 32118 df-sitg 32197 df-sitm 32198

This theorem is referenced by: (None)

W3C validator