rrxnm - Metamath Proof Explorer

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Theorem rrxnm 25380

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 21600	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
2		srngring 20822	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
3	1, 2	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
4		eqid 2741	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
5	4	frlmlmod 21728	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
6	3, 5	mpan 697	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
7		lmodgrp 20861	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
8		eqid 2741	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
9		eqid 2741	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
10		eqid 2741	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
11		eqid 2741	. . . 4 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
12	8, 9, 10, 11	tchnmfval 25217	. . 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 25376	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
16	15	fveq2d 6835	. 2 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
17	15	fveq2d 6835	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
18		rrxbase.b	. . . 4 ⊢ 𝐵 = (Base‘𝐻)
19	8, 10	tcphbas 25208	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
20	17, 18, 19	3eqtr4g 2801	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
21	14, 18	rrxbase 25377	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0})
22		ssrab2 4014	. . . . . . . 8 ⊢ {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} ⊆ (ℝ ↑_m 𝐼)
23	21, 22	eqsstrdi 3961	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ⊆ (ℝ ↑_m 𝐼))
24	23	sselda 3917	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑_m 𝐼))
25	15	fveq2d 6835	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (·_𝑖‘𝐻) = (·_𝑖‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
26	14, 18	rrxip 25379	. . . . . . . . 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 25214	. . . . . . . . . 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 2787	. . . . . . . 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 482	. . . . . . 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 777	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → ℎ = 𝑓)
32	31	fveq1d 6833	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (ℎ‘𝑥) = (𝑓‘𝑥))
33		simprr 779	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → 𝑔 = 𝑓)
34	33	fveq1d 6833	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (𝑔‘𝑥) = (𝑓‘𝑥))
35	32, 34	oveq12d 7378	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
36	35	adantr 482	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
37		elmapi 8790	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℝ)
38	37	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℝ)
39	38	ffvelcdmda 7029	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
40	39	recnd 11168	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
41	40	adantlr 722	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
42	41	sqvald 14100	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)↑2) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
43	36, 42	eqtr4d 2779	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥)↑2))
44	43	mpteq2dva 5168	. . . . . . . 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 7376	. . . . . . 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 486	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → 𝑓 ∈ (ℝ ↑_m 𝐼))
47		ovexd 7395	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))) ∈ V)
48	30, 45, 46, 46, 47	ovmpod 7512	. . . . . 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 598	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))
50	49	eqcomd 2747	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))) = (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓))
51	50	fveq2d 6835	. . 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 5161	. 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 2787	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 397 = wceq 1548 ∈ wcel 2121 {crab 3393 Vcvv 3433 class class class wbr 5075 ↦ cmpt 5156 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 ∈ cmpo 7362 ↑_m cmap 8767 finSupp cfsupp 9268 ℂcc 11031 ℝcr 11032 0cc0 11033 · cmul 11038 2c2 12231 ↑cexp 14018 √csqrt 15190 Basecbs 17174 ·_𝑖cip 17220 Σ_g cgsu 17398 Grpcgrp 18904 Ringcrg 20209 *-Ringcsr 20814 LModclmod 20854 ℝ_fldcrefld 21583 freeLMod cfrlm 21725 normcnm 24563 toℂPreHilctcph 25156 ℝ^crrx 25372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-prds 17405 df-pws 17407 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19094 df-ghm 19183 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-rhm 20447 df-subrng 20522 df-subrg 20546 df-drng 20707 df-field 20708 df-staf 20815 df-srng 20816 df-lmod 20856 df-lss 20926 df-sra 21167 df-rgmod 21168 df-cnfld 21352 df-refld 21584 df-dsmm 21711 df-frlm 21726 df-nm 24569 df-tng 24571 df-tcph 25158 df-rrx 25374

This theorem is referenced by: rrxds 25382

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