Theorem rrxsca 25678

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 2760	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
3	1, 2	rrxprds 25671	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
4	3	fveq2d 6877	. 2 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))))
5		fvex 6886	. . . . 5 ⊢ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ∈ V
6	5	mptex 7217	. . . 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 2760	. . . . . 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 2760	. . . . . 6 ⊢ (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
9	7, 8	tngsca 24925	. . . . 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 2766	. . . 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 14	. . 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 2760	. . . . . 6 ⊢ (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
13		eqid 2760	. . . . . 6 ⊢ (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
14		eqid 2760	. . . . . 6 ⊢ (·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
15	12, 13, 14	tcphval 25500	. . . . 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 6876	. . . 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 2760	. . . . 5 ⊢ (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) = (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))
19		refld 21886	. . . . . 6 ⊢ ℝfld ∈ Field
20	19	a1i 11	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → ℝfld ∈ Field)
21		id 23	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉)
22		snex 5396	. . . . . . 7 ⊢ {((subringAlg ‘ℝfld)‘ℝ)} ∈ V
23	22	a1i 11	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {((subringAlg ‘ℝfld)‘ℝ)} ∈ V)
24	21, 23	xpexd 7748	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}) ∈ V)
25	18, 20, 24	prdssca 17588	. . . 4 ⊢ (𝐼 ∈ 𝑉 → ℝfld = (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))))
26		fvex 6886	. . . . 5 ⊢ (Base‘𝐻) ∈ V
27		eqid 2760	. . . . . 6 ⊢ ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) = ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))
28		eqid 2760	. . . . . 6 ⊢ (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})))
29	27, 28	resssca 17475	. . . . 5 ⊢ ((Base‘𝐻) ∈ V → (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
30	26, 29	mp1i 14	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
31	25, 30	eqtrd 2795	. . 3 ⊢ (𝐼 ∈ 𝑉 → ℝfld = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
32	11, 17, 31	3eqtr4d 2805	. 2 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))) = ℝfld)
33	4, 32	eqtrd 2795	1 ⊢ (𝐼 ∈ 𝑉 → (Scalar‘𝐻) = ℝfld)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450  {csn 4583   ↦ cmpt 5185   × cxp 5645  ‘cfv 6527  (class class class)co 7408  ℝcr 11170  √csqrt 15367  Basecbs 17348   ↾_s cress 17369  Scalarcsca 17392  ·_𝑖cip 17394  X_scprds 17577  Fieldcfield 20942  subringAlg csra 21407  ℝ_fldcrefld 21871   toNrmGrp ctng 24858  toℂPreHilctcph 25449  ℝ^crrx 25665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248  ax-pre-sup 11249  ax-addf 11250

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-map 8827  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-sup 9412  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-div 11943  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-9 12381  df-n0 12576  df-z 12663  df-dec 12784  df-uz 12935  df-rp 13090  df-fz 13609  df-seq 14113  df-exp 14173  df-cj 15233  df-re 15234  df-im 15235  df-sqrt 15369  df-abs 15370  df-struct 17286  df-sets 17303  df-slot 17321  df-ndx 17333  df-base 17349  df-ress 17370  df-plusg 17402  df-mulr 17403  df-starv 17404  df-sca 17405  df-vsca 17406  df-ip 17407  df-tset 17408  df-ple 17409  df-ds 17411  df-unif 17412  df-hom 17413  df-cco 17414  df-0g 17573  df-prds 17579  df-pws 17581  df-mgm 18777  df-sgrp 18869  df-mnd 18885  df-grp 19108  df-minusg 19109  df-subg 19294  df-cmn 19957  df-abl 19958  df-mgp 20322  df-rng 20336  df-ur 20369  df-ring 20422  df-cring 20423  df-oppr 20528  df-dvdsr 20548  df-unit 20549  df-invr 20579  df-dvr 20592  df-subrng 20759  df-subrg 20783  df-drng 20943  df-field 20944  df-sra 21409  df-rgmod 21410  df-cnfld 21640  df-refld 21872  df-dsmm 21999  df-frlm 22014  df-tng 24864  df-tcph 25451  df-rrx 25667

This theorem is used by:  rrxlines  49767