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Theorem rrxmfval 25635

Description: The value of the Euclidean metric. Compare with rrnval 38578. (Contributed by Thierry Arnoux, 30-Jun-2019.)

**Hypotheses**
Ref	Expression
rrxmval.1	⊢ 𝑋 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
rrxmval.d	⊢ 𝐷 = (dist‘(ℝ^‘𝐼))

**Assertion**
Ref	Expression
rrxmfval	⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))

Distinct variable groups: 𝐷,𝑓,𝑔 𝑓,ℎ,𝑘,𝑔,𝐼 𝑓,𝑉,𝑔,ℎ,𝑘 𝑓,𝑋,𝑔,𝑘 Allowed substitution hints: 𝐷(ℎ, 𝑘) 𝑋(ℎ)

Proof of Theorem rrxmfval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . . 5 ⊢ (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))))
2		fvex 6892	. . . . 5 ⊢ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))) ∈ V
3	1, 2	fnmpoi 8068	. . . 4 ⊢ (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) Fn ((Base‘(ℝ^‘𝐼)) × (Base‘(ℝ^‘𝐼)))
4		rrxmval.d	. . . . . 6 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
5		eqid 2760	. . . . . . 7 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
6		eqid 2760	. . . . . . 7 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
7	5, 6	rrxds 25622	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘(ℝ^‘𝐼)))
8	4, 7	eqtr4id 2814	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
9		rrxmval.1	. . . . . . 7 ⊢ 𝑋 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
10	5, 6	rrxbase 25617	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ^‘𝐼)) = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
11	9, 10	eqtr4id 2814	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝑋 = (Base‘(ℝ^‘𝐼)))
12	11	sqxpeqd 5687	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑋 × 𝑋) = ((Base‘(ℝ^‘𝐼)) × (Base‘(ℝ^‘𝐼))))
13	8, 12	fneq12d 6628	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐷 Fn (𝑋 × 𝑋) ↔ (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) Fn ((Base‘(ℝ^‘𝐼)) × (Base‘(ℝ^‘𝐼)))))
14	3, 13	mpbiri 261	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐷 Fn (𝑋 × 𝑋))
15		fnov 7545	. . 3 ⊢ (𝐷 Fn (𝑋 × 𝑋) ↔ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔)))
16	14, 15	sylib 221	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔)))
17	9, 4	rrxmval 25634	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓𝐷𝑔) = (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
18	17	mpoeq3dva 7491	. 2 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔)) = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))
19	16, 18	eqtrd 2795	1 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 {crab 3412 ∪ cun 3897 class class class wbr 5103 ↦ cmpt 5186 × cxp 5653 Fn wfn 6528 ‘cfv 6533 (class class class)co 7414 ∈ cmpo 7416 supp csupp 8159 ↑_m cmap 8827 finSupp cfsupp 9332 ℝcr 11124 0cc0 11125 − cmin 11466 2c2 12320 ↑cexp 14126 √csqrt 15321 Σcsu 15774 Basecbs 17302 distcds 17352 Σ_g cgsu 17526 ℝ_fldcrefld 21818 ℝ^crrx 25612

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204 ax-mulf 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-er 8697 df-map 8829 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-sup 9413 df-oi 9483 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-rp 13044 df-fz 13563 df-fzo 13711 df-seq 14067 df-exp 14127 df-hash 14396 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-clim 15576 df-sum 15775 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-starv 17358 df-sca 17359 df-vsca 17360 df-ip 17361 df-tset 17362 df-ple 17363 df-ds 17365 df-unif 17366 df-hom 17367 df-cco 17368 df-0g 17527 df-gsum 17528 df-prds 17533 df-pws 17535 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-mhm 18892 df-grp 19061 df-minusg 19062 df-sbg 19063 df-subg 19247 df-ghm 19342 df-cntz 19445 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-ring 20375 df-cring 20376 df-oppr 20479 df-dvdsr 20499 df-unit 20500 df-invr 20530 df-dvr 20543 df-rhm 20614 df-subrng 20709 df-subrg 20733 df-drng 20893 df-field 20894 df-staf 21006 df-srng 21007 df-lmod 21047 df-lss 21117 df-sra 21358 df-rgmod 21359 df-cnfld 21587 df-refld 21819 df-dsmm 21946 df-frlm 21961 df-nm 24809 df-tng 24811 df-tcph 25398 df-rrx 25614

This theorem is used by: rrxmet 25637

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