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Theorem rrxsca 24003
 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 (𝐼𝑉 → (Scalar‘𝐻) = ℝfld)

Proof of Theorem rrxsca
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rrxsca.r . . . 4 𝐻 = (ℝ^‘𝐼)
2 eqid 2824 . . . 4 (Base‘𝐻) = (Base‘𝐻)
31, 2rrxprds 23996 . . 3 (𝐼𝑉𝐻 = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
43fveq2d 6665 . 2 (𝐼𝑉 → (Scalar‘𝐻) = (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))))
5 fvex 6674 . . . . 5 (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) ∈ V
65mptex 6977 . . . 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 2824 . . . . . 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 2824 . . . . . 6 (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
97, 8tngsca 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))))))
109eqcomd 2830 . . . 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))))
116, 10mp1i 13 . . 3 (𝐼𝑉 → (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 2824 . . . . . 6 (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
13 eqid 2824 . . . . . 6 (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (Base‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
14 eqid 2824 . . . . . 6 (·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))) = (·𝑖‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))
1512, 13, 14tcphval 23825 . . . . 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))))
1615fveq2i 6664 . . . 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)))))
1716a1i 11 . . 3 (𝐼𝑉 → (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 2824 . . . . 5 (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) = (ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))
19 refld 20763 . . . . . 6 fld ∈ Field
2019a1i 11 . . . . 5 (𝐼𝑉 → ℝfld ∈ Field)
21 id 22 . . . . . 6 (𝐼𝑉𝐼𝑉)
22 snex 5319 . . . . . . 7 {((subringAlg ‘ℝfld)‘ℝ)} ∈ V
2322a1i 11 . . . . . 6 (𝐼𝑉 → {((subringAlg ‘ℝfld)‘ℝ)} ∈ V)
2421, 23xpexd 7468 . . . . 5 (𝐼𝑉 → (𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}) ∈ V)
2518, 20, 24prdssca 16729 . . . 4 (𝐼𝑉 → ℝfld = (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))))
26 fvex 6674 . . . . 5 (Base‘𝐻) ∈ V
27 eqid 2824 . . . . . 6 ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)) = ((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))
28 eqid 2824 . . . . . 6 (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})))
2927, 28resssca 16650 . . . . 5 ((Base‘𝐻) ∈ V → (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
3026, 29mp1i 13 . . . 4 (𝐼𝑉 → (Scalar‘(ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)}))) = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
3125, 30eqtrd 2859 . . 3 (𝐼𝑉 → ℝfld = (Scalar‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻))))
3211, 17, 313eqtr4d 2869 . 2 (𝐼𝑉 → (Scalar‘(toℂPreHil‘((ℝfldXs(𝐼 × {((subringAlg ‘ℝfld)‘ℝ)})) ↾s (Base‘𝐻)))) = ℝfld)
334, 32eqtrd 2859 1 (𝐼𝑉 → (Scalar‘𝐻) = ℝfld)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  Vcvv 3480  {csn 4550   ↦ cmpt 5132   × cxp 5540  ‘cfv 6343  (class class class)co 7149  ℝcr 10534  √csqrt 14592  Basecbs 16483   ↾s cress 16484  Scalarcsca 16568  ·𝑖cip 16570  Xscprds 16719  Fieldcfield 19503  subringAlg csra 19940  ℝfldcrefld 20748   toNrmGrp ctng 23188  toℂPreHilctcph 23775  ℝ^crrx 23990 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613  ax-addf 10614  ax-mulf 10615 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-tpos 7888  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-rp 12387  df-fz 12895  df-seq 13374  df-exp 13435  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 19376  df-dvdsr 19394  df-unit 19395  df-invr 19425  df-dvr 19436  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 23777  df-rrx 23992 This theorem is referenced by:  rrxlines  45073
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