rrxnm - Metamath Proof Explorer

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Theorem rrxnm 25351

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 21580	. . . . 5 ⊢ ℝ_fld ∈ *-Ring
2		srngring 20783	. . . . 5 ⊢ (ℝ_fld ∈ *-Ring → ℝ_fld ∈ Ring)
3	1, 2	ax-mp 5	. . . 4 ⊢ ℝ_fld ∈ Ring
4		eqid 2737	. . . . 5 ⊢ (ℝ_fld freeLMod 𝐼) = (ℝ_fld freeLMod 𝐼)
5	4	frlmlmod 21708	. . . 4 ⊢ ((ℝ_fld ∈ Ring ∧ 𝐼 ∈ 𝑉) → (ℝ_fld freeLMod 𝐼) ∈ LMod)
6	3, 5	mpan 691	. . 3 ⊢ (𝐼 ∈ 𝑉 → (ℝ_fld freeLMod 𝐼) ∈ LMod)
7		lmodgrp 20822	. . 3 ⊢ ((ℝ_fld freeLMod 𝐼) ∈ LMod → (ℝ_fld freeLMod 𝐼) ∈ Grp)
8		eqid 2737	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼))
9		eqid 2737	. . . 4 ⊢ (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
10		eqid 2737	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(ℝ_fld freeLMod 𝐼))
11		eqid 2737	. . . 4 ⊢ (·_𝑖‘(ℝ_fld freeLMod 𝐼)) = (·_𝑖‘(ℝ_fld freeLMod 𝐼))
12	8, 9, 10, 11	tchnmfval 25188	. . 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 25347	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
16	15	fveq2d 6839	. 2 ⊢ (𝐼 ∈ 𝑉 → (norm‘𝐻) = (norm‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
17	15	fveq2d 6839	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝐻) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
18		rrxbase.b	. . . 4 ⊢ 𝐵 = (Base‘𝐻)
19	8, 10	tcphbas 25179	. . . 4 ⊢ (Base‘(ℝ_fld freeLMod 𝐼)) = (Base‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
20	17, 18, 19	3eqtr4g 2797	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐵 = (Base‘(ℝ_fld freeLMod 𝐼)))
21	14, 18	rrxbase 25348	. . . . . . . 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 3979	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐵 ⊆ (ℝ ↑_m 𝐼))
24	23	sselda 3934	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (ℝ ↑_m 𝐼))
25	15	fveq2d 6839	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → (·_𝑖‘𝐻) = (·_𝑖‘(toℂPreHil‘(ℝ_fld freeLMod 𝐼))))
26	14, 18	rrxip 25350	. . . . . . . . 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 25185	. . . . . . . . . 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 2783	. . . . . . . 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 771	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → ℎ = 𝑓)
32	31	fveq1d 6837	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → (ℎ‘𝑥) = (𝑓‘𝑥))
33		simprr 773	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) → 𝑔 = 𝑓)
34	33	fveq1d 6837	. . . . . . . . . . . 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 480	. . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) → 𝑓:𝐼⟶ℝ)
39	38	ffvelcdmda 7031	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℝ)
40	39	recnd 11164	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
41	40	adantlr 716	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ ℂ)
42	41	sqvald 14070	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((𝑓‘𝑥)↑2) = ((𝑓‘𝑥) · (𝑓‘𝑥)))
43	36, 42	eqtr4d 2775	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ (ℝ ↑_m 𝐼)) ∧ (ℎ = 𝑓 ∧ 𝑔 = 𝑓)) ∧ 𝑥 ∈ 𝐼) → ((ℎ‘𝑥) · (𝑔‘𝑥)) = ((𝑓‘𝑥)↑2))
44	43	mpteq2dva 5192	. . . . . . . 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 484	. . . . . . 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 592	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))))
50	49	eqcomd 2743	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑓 ∈ 𝐵) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)↑2))) = (𝑓(·_𝑖‘(ℝ_fld freeLMod 𝐼))𝑓))
51	50	fveq2d 6839	. . 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 5185	. 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 2783	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 1542 ∈ wcel 2114 {crab 3400 Vcvv 3441 class class class wbr 5099 ↦ cmpt 5180 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ∈ cmpo 7362 ↑_m cmap 8767 finSupp cfsupp 9268 ℂcc 11028 ℝcr 11029 0cc0 11030 · cmul 11035 2c2 12204 ↑cexp 13988 √csqrt 15160 Basecbs 17140 ·_𝑖cip 17186 Σ_g cgsu 17364 Grpcgrp 18867 Ringcrg 20172 *-Ringcsr 20775 LModclmod 20815 ℝ_fldcrefld 21563 freeLMod cfrlm 21705 normcnm 24524 toℂPreHilctcph 25127 ℝ^crrx 25343

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 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 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-rp 12910 df-fz 13428 df-seq 13929 df-exp 13989 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-0g 17365 df-prds 17371 df-pws 17373 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-grp 18870 df-minusg 18871 df-sbg 18872 df-subg 19057 df-ghm 19146 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-dvr 20341 df-rhm 20412 df-subrng 20483 df-subrg 20507 df-drng 20668 df-field 20669 df-staf 20776 df-srng 20777 df-lmod 20817 df-lss 20887 df-sra 21129 df-rgmod 21130 df-cnfld 21314 df-refld 21564 df-dsmm 21691 df-frlm 21706 df-nm 24530 df-tng 24532 df-tcph 25129 df-rrx 25345

This theorem is referenced by: rrxds 25353

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