Theorem rrxsca 23999

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 2821	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
3	1, 2	rrxprds 23992	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
4	3	fveq2d 6674	. 2 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))))
5		fvex 6683	. . . . 5 ⊢ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ∈ V
6	5	mptex 6986	. . . 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 2821	. . . . . 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 2821	. . . . . 6 ⊢ (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
9	7, 8	tngsca 23254	. . . . 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 2827	. . . 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 2821	. . . . . 6 ⊢ (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
13		eqid 2821	. . . . . 6 ⊢ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
14		eqid 2821	. . . . . 6 ⊢ (·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
15	12, 13, 14	tcphval 23821	. . . . 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 6673	. . . 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 2821	. . . . 5 ⊢ (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) = (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))
19		refld 20763	. . . . . 6 ⊢ ℝfld ∈ Field
20	19	a1i 11	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → ℝfld ∈ Field)
21		id 22	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉)
22		snex 5332	. . . . . . 7 ⊢ {((subringAlg ‘ℝfld)‘ℝ)} ∈ V
23	22	a1i 11	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {((subringAlg ‘ℝfld)‘ℝ)} ∈ V)
24	21, 23	xpexd 7474	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}) ∈ V)
25	18, 20, 24	prdssca 16729	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝfld = (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))))
26		fvex 6683	. . . . 5 ⊢ (Base‘𝐻) ∈ V
27		eqid 2821	. . . . . 6 ⊢ ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) = ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))
28		eqid 2821	. . . . . 6 ⊢ (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})))
29	27, 28	resssca 16650	. . . . 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 2856	. . 3 ⊢ (𝐼 ∈ 𝑉 → ℝfld = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
32	11, 17, 31	3eqtr4d 2866	. 2 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))) = ℝfld)
33	4, 32	eqtrd 2856	1 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = ℝfld)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3494  {csn 4567   ↦ cmpt 5146   × cxp 5553  ‘cfv 6355  (class class class)co 7156  ℝcr 10536  √csqrt 14592  Basecbs 16483   ↾_s cress 16484  Scalarcsca 16568  ·_𝑖cip 16570  X_scprds 16719  Fieldcfield 19503  subringAlg csra 19940  ℝ_fldcrefld 20748   toNrmGrp ctng 23188  toℂPreHilctcph 23771  ℝ^crrx 23986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615  ax-addf 10616  ax-mulf 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-tpos 7892  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-rp 12391  df-fz 12894  df-seq 13371  df-exp 13431  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-hom 16589  df-cco 16590  df-0g 16715  df-prds 16721  df-pws 16723  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-subg 18276  df-cmn 18908  df-mgp 19240  df-ur 19252  df-ring 19299  df-cring 19300  df-oppr 19373  df-dvdsr 19391  df-unit 19392  df-invr 19422  df-dvr 19433  df-drng 19504  df-field 19505  df-subrg 19533  df-sra 19944  df-rgmod 19945  df-cnfld 20546  df-refld 20749  df-dsmm 20876  df-frlm 20891  df-tng 23194  df-tcph 23773  df-rrx 23988

This theorem is referenced by:  rrxlines  44769