Theorem rrxsca 25444

Description: The field of real numbers is the scalar field of the generalized real Euclidean space. (Contributed by AV, 15-Jan-2023.)

Hypothesis

Ref	Expression
rrxsca.r	⊢ 𝐻 = (ℝ^‘𝐼)

Assertion

Ref	Expression
rrxsca	⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = ℝfld)

Proof of Theorem rrxsca

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxsca.r	. . . 4 ⊢ 𝐻 = (ℝ^‘𝐼)
2		eqid 2735	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
3	1, 2	rrxprds 25437	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
4	3	fveq2d 6911	. 2 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))))
5		fvex 6920	. . . . 5 ⊢ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ∈ V
6	5	mptex 7243	. . . 4 ⊢ (𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥))) ∈ V
7		eqid 2735	. . . . . 6 ⊢ (((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) toNrmGrp (𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥)))) = (((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) toNrmGrp (𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥))))
8		eqid 2735	. . . . . 6 ⊢ (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
9	7, 8	tngsca 24678	. . . . 5 ⊢ ((𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥))) ∈ V → (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (Scalar‘(((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) toNrmGrp (𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥))))))
10	9	eqcomd 2741	. . . 4 ⊢ ((𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥))) ∈ V → (Scalar‘(((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) toNrmGrp (𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥))))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
11	6, 10	mp1i 13	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) toNrmGrp (𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥))))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
12		eqid 2735	. . . . . 6 ⊢ (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
13		eqid 2735	. . . . . 6 ⊢ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
14		eqid 2735	. . . . . 6 ⊢ (·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
15	12, 13, 14	tcphval 25266	. . . . 5 ⊢ (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) toNrmGrp (𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥))))
16	15	fveq2i 6910	. . . 4 ⊢ (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))) = (Scalar‘(((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) toNrmGrp (𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥)))))
17	16	a1i 11	. . 3 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))) = (Scalar‘(((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) toNrmGrp (𝑥 ∈ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ↦ (√‘(𝑥(·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))𝑥))))))
18		eqid 2735	. . . . 5 ⊢ (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) = (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))
19		refld 21655	. . . . . 6 ⊢ ℝfld ∈ Field
20	19	a1i 11	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → ℝfld ∈ Field)
21		id 22	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉)
22		snex 5442	. . . . . . 7 ⊢ {((subringAlg ‘ℝfld)‘ℝ)} ∈ V
23	22	a1i 11	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {((subringAlg ‘ℝfld)‘ℝ)} ∈ V)
24	21, 23	xpexd 7770	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}) ∈ V)
25	18, 20, 24	prdssca 17503	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝfld = (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))))
26		fvex 6920	. . . . 5 ⊢ (Base‘𝐻) ∈ V
27		eqid 2735	. . . . . 6 ⊢ ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) = ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))
28		eqid 2735	. . . . . 6 ⊢ (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})))
29	27, 28	resssca 17389	. . . . 5 ⊢ ((Base‘𝐻) ∈ V → (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
30	26, 29	mp1i 13	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
31	25, 30	eqtrd 2775	. . 3 ⊢ (𝐼 ∈ 𝑉 → ℝfld = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
32	11, 17, 31	3eqtr4d 2785	. 2 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))) = ℝfld)
33	4, 32	eqtrd 2775	1 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = ℝfld)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631   ↦ cmpt 5231   × cxp 5687  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  √csqrt 15269  Basecbs 17245   ↾_s cress 17274  Scalarcsca 17301  ·_𝑖cip 17303  X_scprds 17492  Fieldcfield 20747  subringAlg csra 21188  ℝ_fldcrefld 21640   toNrmGrp ctng 24607  toℂPreHilctcph 25215  ℝ^crrx 25431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  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 17488  df-prds 17494  df-pws 17496  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-subrng 20563  df-subrg 20587  df-drng 20748  df-field 20749  df-sra 21190  df-rgmod 21191  df-cnfld 21383  df-refld 21641  df-dsmm 21770  df-frlm 21785  df-tng 24613  df-tcph 25217  df-rrx 25433

This theorem is referenced by:  rrxlines  48583