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Theorem rrxmfval 25440

Description: The value of the Euclidean metric. Compare with rrnval 37834. (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 2737	. . . . 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 6919	. . . . 5 ⊢ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))) ∈ V
3	1, 2	fnmpoi 8095	. . . 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 2737	. . . . . . 7 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
6		eqid 2737	. . . . . . 7 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
7	5, 6	rrxds 25427	. . . . . 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 2796	. . . . 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 25422	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ^‘𝐼)) = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
11	9, 10	eqtr4id 2796	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝑋 = (Base‘(ℝ^‘𝐼)))
12	11	sqxpeqd 5717	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑋 × 𝑋) = ((Base‘(ℝ^‘𝐼)) × (Base‘(ℝ^‘𝐼))))
13	8, 12	fneq12d 6663	. . . 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 258	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐷 Fn (𝑋 × 𝑋))
15		fnov 7564	. . 3 ⊢ (𝐷 Fn (𝑋 × 𝑋) ↔ 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔)))
16	14, 15	sylib 218	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (𝑓𝐷𝑔)))
17	9, 4	rrxmval 25439	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑓𝐷𝑔) = (√‘Σ𝑘 ∈ ((𝑓 supp 0) ∪ (𝑔 supp 0))(((𝑓‘𝑘) − (𝑔‘𝑘))↑2)))
18	17	mpoeq3dva 7510	. 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 2777	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 1540 ∈ wcel 2108 {crab 3436 ∪ cun 3949 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 supp csupp 8185 ↑_m cmap 8866 finSupp cfsupp 9401 ℝcr 11154 0cc0 11155 − cmin 11492 2c2 12321 ↑cexp 14102 √csqrt 15272 Σcsu 15722 Basecbs 17247 distcds 17306 Σ_g cgsu 17485 ℝ_fldcrefld 21622 ℝ^crrx 25417

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-field 20732 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-refld 21623 df-dsmm 21752 df-frlm 21767 df-nm 24595 df-tng 24597 df-tcph 25203 df-rrx 25419

This theorem is referenced by: rrxmet 25442

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