rrxnm - Metamath Proof Explorer

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Theorem rrxnm 25561

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 21782	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
2		srngring 20960	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
3	1, 2	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
4		eqid 2762	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
5	4	frlmlmod 21910	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
6	3, 5	mpan 702	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
7		lmodgrp 20999	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
8		eqid 2762	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
9		eqid 2762	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
10		eqid 2762	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
11		eqid 2762	. . . 4 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
12	8, 9, 10, 11	tchnmfval 25398	. . 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 19	. 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 25557	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
16	15	fveq2d 6885	. 2 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
17	15	fveq2d 6885	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
18		rrxbase.b	. . . 4 ⊢ 𝐵 = (Base‘𝐻)
19	8, 10	tcphbas 25389	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
20	17, 18, 19	3eqtr4g 2822	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
21	14, 18	rrxbase 25558	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0})
22		ssrab2 4033	. . . . . . . 8 ⊢ {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} ⊆ (ℝ ↑_m 𝐼)
23	21, 22	eqsstrdi 3980	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ⊆ (ℝ ↑_m 𝐼))
24	23	sselda 3936	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑_m 𝐼))
25	15	fveq2d 6885	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (·_𝑖‘𝐻) = (·_𝑖‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
26	14, 18	rrxip 25560	. . . . . . . . 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 25395	. . . . . . . . . 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 2808	. . . . . . . 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 485	. . . . . . 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 782	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → ℎ = 𝑓)
32	31	fveq1d 6883	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (ℎ‘𝑥) = (𝑓‘𝑥))
33		simprr 784	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → 𝑔 = 𝑓)
34	33	fveq1d 6883	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (𝑔‘𝑥) = (𝑓‘𝑥))
35	32, 34	oveq12d 7430	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
36	35	adantr 485	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
37		elmapi 8844	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℝ)
38	37	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℝ)
39	38	ffvelcdmda 7079	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
40	39	recnd 11243	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
41	40	adantlr 727	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
42	41	sqvald 14186	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)↑2) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
43	36, 42	eqtr4d 2800	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥)↑2))
44	43	mpteq2dva 5203	. . . . . . . 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 7428	. . . . . . 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 489	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → 𝑓 ∈ (ℝ ↑_m 𝐼))
47		ovexd 7447	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))) ∈ V)
48	30, 45, 46, 46, 47	ovmpod 7564	. . . . . 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 602	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))
50	49	eqcomd 2768	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))) = (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓))
51	50	fveq2d 6885	. . 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 5196	. 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 2808	1 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝐵 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))) = (norm‘𝐻))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 {crab 3415 Vcvv 3454 class class class wbr 5108 ↦ cmpt 5191 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 ∈ cmpo 7414 ↑_m cmap 8822 finSupp cfsupp 9319 ℂcc 11104 ℝcr 11105 0cc0 11106 · cmul 11111 2c2 12301 ↑cexp 14104 √csqrt 15291 Basecbs 17275 ·_𝑖cip 17321 Σ_g cgsu 17499 Grpcgrp 19006 Ringcrg 20321 *-Ringcsr 20952 LModclmod 20992 ℝ_fldcrefld 21765 freeLMod cfrlm 21907 normcnm 24744 toℂPreHilctcph 25337 ℝ^crrx 25553

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-sup 9400 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-rp 13023 df-fz 13542 df-seq 14045 df-exp 14105 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-0g 17500 df-prds 17506 df-pws 17508 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-grp 19009 df-minusg 19010 df-sbg 19011 df-subg 19195 df-ghm 19290 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-invr 20477 df-dvr 20490 df-rhm 20561 df-subrng 20656 df-subrg 20680 df-drng 20840 df-field 20841 df-staf 20953 df-srng 20954 df-lmod 20994 df-lss 21064 df-sra 21305 df-rgmod 21306 df-cnfld 21534 df-refld 21766 df-dsmm 21893 df-frlm 21908 df-nm 24750 df-tng 24752 df-tcph 25339 df-rrx 25555

This theorem is used by: rrxds 25563

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