rrxnm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rrxnm		Structured version Visualization version GIF version

Theorem rrxnm 25426

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 21640	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
2		srngring 20848	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
3	1, 2	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
4		eqid 2736	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
5	4	frlmlmod 21770	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
6	3, 5	mpan 690	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
7		lmodgrp 20866	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
8		eqid 2736	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
9		eqid 2736	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
10		eqid 2736	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
11		eqid 2736	. . . 4 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
12	8, 9, 10, 11	tchnmfval 25263	. . 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 25422	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
16	15	fveq2d 6909	. 2 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
17	15	fveq2d 6909	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
18		rrxbase.b	. . . 4 ⊢ 𝐵 = (Base‘𝐻)
19	8, 10	tcphbas 25254	. . . 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 25423	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0})
22		ssrab2 4079	. . . . . . . 8 ⊢ {𝑓 ∈ (ℝ ↑_m 𝐼) ∣ 𝑓 finSupp 0} ⊆ (ℝ ↑_m 𝐼)
23	21, 22	eqsstrdi 4027	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ⊆ (ℝ ↑_m 𝐼))
24	23	sselda 3982	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑_m 𝐼))
25	15	fveq2d 6909	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (·_𝑖‘𝐻) = (·_𝑖‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
26	14, 18	rrxip 25425	. . . . . . . . 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 25260	. . . . . . . . . 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 480	. . . . . . 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 770	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → ℎ = 𝑓)
32	31	fveq1d 6907	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (ℎ‘𝑥) = (𝑓‘𝑥))
33		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → 𝑔 = 𝑓)
34	33	fveq1d 6907	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (𝑔‘𝑥) = (𝑓‘𝑥))
35	32, 34	oveq12d 7450	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
36	35	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
37		elmapi 8890	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (ℝ ↑_m 𝐼) → 𝑓:𝐼⟶ℝ)
38	37	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℝ)
39	38	ffvelcdmda 7103	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
40	39	recnd 11290	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
41	40	adantlr 715	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
42	41	sqvald 14184	. . . . . . . . . 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 5241	. . . . . . . 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 7448	. . . . . . 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 484	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → 𝑓 ∈ (ℝ ↑_m 𝐼))
47		ovexd 7467	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))) ∈ V)
48	30, 45, 46, 46, 47	ovmpod 7586	. . . . . 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 591	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))
50	49	eqcomd 2742	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))) = (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓))
51	50	fveq2d 6909	. . 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 5230	. 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 395 = wceq 1539 ∈ wcel 2107 {crab 3435 Vcvv 3479 class class class wbr 5142 ↦ cmpt 5224 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 ↑_m cmap 8867 finSupp cfsupp 9402 ℂcc 11154 ℝcr 11155 0cc0 11156 · cmul 11161 2c2 12322 ↑cexp 14103 √csqrt 15273 Basecbs 17248 ·_𝑖cip 17303 Σ_g cgsu 17486 Grpcgrp 18952 Ringcrg 20231 *-Ringcsr 20840 LModclmod 20859 ℝ_fldcrefld 21623 freeLMod cfrlm 21767 normcnm 24590 toℂPreHilctcph 25202 ℝ^crrx 25418

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-rp 13036 df-fz 13549 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17487 df-prds 17493 df-pws 17495 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-ghm 19232 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-drng 20732 df-field 20733 df-staf 20841 df-srng 20842 df-lmod 20861 df-lss 20931 df-sra 21173 df-rgmod 21174 df-cnfld 21366 df-refld 21624 df-dsmm 21753 df-frlm 21768 df-nm 24596 df-tng 24598 df-tcph 25204 df-rrx 25420

This theorem is referenced by: rrxds 25428

W3C validator