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Theorem rrxmfval 25362

Description: The value of the Euclidean metric. Compare with rrnval 37341. (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 2728	. . . . 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 6915	. . . . 5 ⊢ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))) ∈ V
3	1, 2	fnmpoi 8082	. . . 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 2728	. . . . . . 7 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
6		eqid 2728	. . . . . . 7 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
7	5, 6	rrxds 25349	. . . . . 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 2787	. . . . 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 25344	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ^‘𝐼)) = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
11	9, 10	eqtr4id 2787	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝑋 = (Base‘(ℝ^‘𝐼)))
12	11	sqxpeqd 5714	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑋 × 𝑋) = ((Base‘(ℝ^‘𝐼)) × (Base‘(ℝ^‘𝐼))))
13	8, 12	fneq12d 6654	. . . 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 257	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐷 Fn (𝑋 × 𝑋))
15		fnov 7559	. . 3 ⊢ (𝐷 Fn (𝑋 × 𝑋) ↔ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔)))
16	14, 15	sylib 217	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔)))
17	9, 4	rrxmval 25361	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓𝐷𝑔) = (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
18	17	mpoeq3dva 7504	. 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 2768	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

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3430 ∪ cun 3947 class class class wbr 5152 ↦ cmpt 5235 × cxp 5680 Fn wfn 6548 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 supp csupp 8173 ↑_m cmap 8853 finSupp cfsupp 9395 ℝcr 11147 0cc0 11148 − cmin 11484 2c2 12307 ↑cexp 14068 √csqrt 15222 Σcsu 15674 Basecbs 17189 distcds 17251 Σ_g cgsu 17431 ℝ_fldcrefld 21550 ℝ^crrx 25339

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 ax-mulf 11228

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-rp 13017 df-fz 13527 df-fzo 13670 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-clim 15474 df-sum 15675 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-0g 17432 df-gsum 17433 df-prds 17438 df-pws 17440 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19092 df-ghm 19182 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-rhm 20425 df-subrng 20497 df-subrg 20522 df-drng 20640 df-field 20641 df-staf 20739 df-srng 20740 df-lmod 20759 df-lss 20830 df-sra 21072 df-rgmod 21073 df-cnfld 21294 df-refld 21551 df-dsmm 21680 df-frlm 21695 df-nm 24519 df-tng 24521 df-tcph 25125 df-rrx 25341

This theorem is referenced by: rrxmet 25364

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