rrxnm - Metamath Proof Explorer

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Theorem rrxnm 25407

Description: The norm of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)

**Hypotheses**
Ref	Expression
rrxval.r	⊢ 𝐻 = (ℝ^‘𝐼)
rrxbase.b	⊢ 𝐵 = (Base‘𝐻)

**Assertion**
Ref	Expression
rrxnm	⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))) = (norm‘𝐻))

Distinct variable groups: 𝑥,𝑓,𝐵 𝑓,𝐼,𝑥 𝑓,𝑉,𝑥 Allowed substitution hints: 𝐻(𝑥,𝑓)

Proof of Theorem rrxnm Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resrng 21613	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
2		srngring 20821	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
3	1, 2	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
4		eqid 2726	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
5	4	frlmlmod 21743	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
6	3, 5	mpan 688	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
7		lmodgrp 20839	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
8		eqid 2726	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
9		eqid 2726	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
10		eqid 2726	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
11		eqid 2726	. . . 4 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
12	8, 9, 10, 11	tchnmfval 25244	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ Grp → (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓))))
13	6, 7, 12	3syl 18	. 2 ⊢ (𝐼 ∈ 𝑉 → (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓))))
14		rrxval.r	. . . 4 ⊢ 𝐻 = (ℝ^‘𝐼)
15	14	rrxval 25403	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
16	15	fveq2d 6897	. 2 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
17	15	fveq2d 6897	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
18		rrxbase.b	. . . 4 ⊢ 𝐵 = (Base‘𝐻)
19	8, 10	tcphbas 25235	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
20	17, 18, 19	3eqtr4g 2791	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
21	14, 18	rrxbase 25404	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0})
22		ssrab2 4073	. . . . . . . 8 ⊢ {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} ⊆ (ℝ ↑_m 𝐼)
23	21, 22	eqsstrdi 4033	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ⊆ (ℝ ↑_m 𝐼))
24	23	sselda 3978	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑_m 𝐼))
25	15	fveq2d 6897	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (·_𝑖‘𝐻) = (·_𝑖‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
26	14, 18	rrxip 25406	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (ℎ ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥))))) = (·_𝑖‘𝐻))
27	8, 11	tcphip 25241	. . . . . . . . . 10 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
28	27	a1i 11	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
29	25, 26, 28	3eqtr4rd 2777	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (ℎ ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥))))))
30	29	adantr 479	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (ℎ ∈ (ℝ ↑_m 𝐼), 𝑔 ∈ (ℝ ↑_m 𝐼) ↦ (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥))))))
31		simprl 769	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → ℎ = 𝑓)
32	31	fveq1d 6895	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (ℎ‘𝑥) = (𝑓‘𝑥))
33		simprr 771	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → 𝑔 = 𝑓)
34	33	fveq1d 6895	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (𝑔‘𝑥) = (𝑓‘𝑥))
35	32, 34	oveq12d 7434	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
36	35	adantr 479	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
37		elmapi 8870	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℝ)
38	37	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℝ)
39	38	ffvelcdmda 7090	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
40	39	recnd 11283	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
41	40	adantlr 713	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
42	41	sqvald 14156	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)↑2) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
43	36, 42	eqtr4d 2769	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥)↑2))
44	43	mpteq2dva 5245	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2)))
45	44	oveq2d 7432	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((ℎ‘𝑥) · (𝑔‘𝑥)))) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))
46		simpr 483	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → 𝑓 ∈ (ℝ ↑_m 𝐼))
47		ovexd 7451	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))) ∈ V)
48	30, 45, 46, 46, 47	ovmpod 7570	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))
49	24, 48	syldan 589	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))
50	49	eqcomd 2732	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))) = (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓))
51	50	fveq2d 6897	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2)))) = (√‘(𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓)))
52	20, 51	mpteq12dva 5234	. 2 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))) = (𝑓 ∈ (Base‘(ℝ_fld freeLMod 𝐼)) ↦ (√‘(𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓))))
53	13, 16, 52	3eqtr4rd 2777	1 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))) = (norm‘𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {crab 3419 Vcvv 3462 class class class wbr 5145 ↦ cmpt 5228 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 ∈ cmpo 7418 ↑_m cmap 8847 finSupp cfsupp 9398 ℂcc 11147 ℝcr 11148 0cc0 11149 · cmul 11154 2c2 12313 ↑cexp 14075 √csqrt 15233 Basecbs 17208 ·_𝑖cip 17266 Σ_g cgsu 17450 Grpcgrp 18923 Ringcrg 20212 *-Ringcsr 20813 LModclmod 20832 ℝ_fldcrefld 21596 freeLMod cfrlm 21740 normcnm 24573 toℂPreHilctcph 25183 ℝ^crrx 25399

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 ax-mulf 11229

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-sup 9478 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-rp 13023 df-fz 13533 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-0g 17451 df-prds 17457 df-pws 17459 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19113 df-ghm 19203 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-rhm 20450 df-subrng 20524 df-subrg 20549 df-drng 20705 df-field 20706 df-staf 20814 df-srng 20815 df-lmod 20834 df-lss 20905 df-sra 21147 df-rgmod 21148 df-cnfld 21340 df-refld 21597 df-dsmm 21726 df-frlm 21741 df-nm 24579 df-tng 24581 df-tcph 25185 df-rrx 25401

This theorem is referenced by: rrxds 25409

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