sitmf - Mathbox for Thierry Arnoux

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Theorem sitmf 34320

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 2729	. . . . . 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 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑓 ∈ dom (𝑊sitg𝑀))
7		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → 𝑔 ∈ dom (𝑊sitg𝑀))
8	1, 3, 5, 6, 7	sitmfval 34318	. . . . 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 34319	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → (𝑓(𝑊sitm𝑀)𝑔) ∈ (0[,]+∞))
12	8, 11	eqeltrrd 2829	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ dom (𝑊sitg𝑀) ∧ 𝑔 ∈ dom (𝑊sitg𝑀))) → (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔)) ∈ (0[,]+∞))
13	12	ralrimivva 3172	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)∀𝑔 ∈ dom (𝑊sitg𝑀)(((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔)) ∈ (0[,]+∞))
14		eqid 2729	. . . 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 8003	. . 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 218	. 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 34317	. . 3 ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ^*_𝑠 ↾_s (0[,]+∞))sitg𝑀)‘(𝑓 ∘_f (dist‘𝑊)𝑔))))
18	17	feq1d 6634	. 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 257	1 ⊢ (𝜑 → (𝑊sitm𝑀):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2109 ∀wral 3044 ∪ cuni 4858 × cxp 5617 dom cdm 5619 ran crn 5620 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 ∘_f cof 7611 0cc0 11009 +∞cpnf 11146 [,]cicc 13251 ↾_s cress 17141 distcds 17170 ℝ^*_𝑠cxrs 17404 Mndcmnd 18608 ∞MetSpcxms 24203 measurescmeas 34162 sitmcsitm 34296 sitgcsitg 34297

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-ac2 10357 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 ax-addf 11088 ax-mulf 11089 ax-xrssca 32958 ax-xrsvsca 32959

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-ac 10010 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-dvds 16164 df-gcd 16406 df-numer 16646 df-denom 16647 df-gz 16842 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-ordt 17405 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-toset 18321 df-ps 18472 df-tsr 18473 df-plusf 18513 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-submnd 18658 df-grp 18815 df-minusg 18816 df-sbg 18817 df-mulg 18947 df-subg 19002 df-ghm 19092 df-cntz 19196 df-od 19407 df-cmn 19661 df-abl 19662 df-omnd 20000 df-ogrp 20001 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-rhm 20357 df-nzr 20398 df-subrng 20431 df-subrg 20455 df-drng 20616 df-field 20617 df-abv 20694 df-orng 20744 df-ofld 20745 df-lmod 20765 df-scaf 20766 df-sra 21077 df-rgmod 21078 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-metu 21260 df-cnfld 21262 df-zring 21354 df-zrh 21410 df-zlm 21411 df-chr 21412 df-refld 21512 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-t1 23199 df-haus 23200 df-reg 23201 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-fcls 23826 df-cnext 23945 df-tmd 23957 df-tgp 23958 df-tsms 24012 df-trg 24045 df-ust 24086 df-utop 24117 df-uss 24142 df-usp 24143 df-ucn 24161 df-cfilu 24172 df-cusp 24183 df-xms 24206 df-ms 24207 df-tms 24208 df-nm 24468 df-ngp 24469 df-nrg 24471 df-nlm 24472 df-ii 24768 df-cncf 24769 df-cfil 25153 df-cmet 25155 df-cms 25233 df-limc 25765 df-dv 25766 df-log 26463 df-qqh 33938 df-rrh 33962 df-rrext 33966 df-esum 33995 df-siga 34076 df-sigagen 34106 df-meas 34163 df-mbfm 34217 df-sitg 34298 df-sitm 34299

This theorem is referenced by: (None)

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