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Theorem rrxsca 24904
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^β€˜πΌ)
Assertion
Ref Expression
rrxsca (𝐼 ∈ 𝑉 β†’ (Scalarβ€˜π») = ℝfld)

Proof of Theorem rrxsca
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 rrxsca.r . . . 4 𝐻 = (ℝ^β€˜πΌ)
2 eqid 2732 . . . 4 (Baseβ€˜π») = (Baseβ€˜π»)
31, 2rrxprds 24897 . . 3 (𝐼 ∈ 𝑉 β†’ 𝐻 = (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))))
43fveq2d 6892 . 2 (𝐼 ∈ 𝑉 β†’ (Scalarβ€˜π») = (Scalarβ€˜(toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))))
5 fvex 6901 . . . . 5 (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ∈ V
65mptex 7221 . . . 4 (π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯))) ∈ V
7 eqid 2732 . . . . . 6 (((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)) toNrmGrp (π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯)))) = (((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)) toNrmGrp (π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯))))
8 eqid 2732 . . . . . 6 (Scalarβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) = (Scalarβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))
97, 8tngsca 24149 . . . . 5 ((π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯))) ∈ V β†’ (Scalarβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) = (Scalarβ€˜(((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)) toNrmGrp (π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯))))))
109eqcomd 2738 . . . 4 ((π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯))) ∈ V β†’ (Scalarβ€˜(((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)) toNrmGrp (π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯))))) = (Scalarβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))))
116, 10mp1i 13 . . 3 (𝐼 ∈ 𝑉 β†’ (Scalarβ€˜(((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)) toNrmGrp (π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯))))) = (Scalarβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))))
12 eqid 2732 . . . . . 6 (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) = (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))
13 eqid 2732 . . . . . 6 (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) = (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))
14 eqid 2732 . . . . . 6 (Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) = (Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))
1512, 13, 14tcphval 24726 . . . . 5 (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) = (((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)) toNrmGrp (π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯))))
1615fveq2i 6891 . . . 4 (Scalarβ€˜(toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))) = (Scalarβ€˜(((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)) toNrmGrp (π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯)))))
1716a1i 11 . . 3 (𝐼 ∈ 𝑉 β†’ (Scalarβ€˜(toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))) = (Scalarβ€˜(((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)) toNrmGrp (π‘₯ ∈ (Baseβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))π‘₯))))))
18 eqid 2732 . . . . 5 (ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) = (ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))
19 refld 21163 . . . . . 6 ℝfld ∈ Field
2019a1i 11 . . . . 5 (𝐼 ∈ 𝑉 β†’ ℝfld ∈ Field)
21 id 22 . . . . . 6 (𝐼 ∈ 𝑉 β†’ 𝐼 ∈ 𝑉)
22 snex 5430 . . . . . . 7 {((subringAlg β€˜β„fld)β€˜β„)} ∈ V
2322a1i 11 . . . . . 6 (𝐼 ∈ 𝑉 β†’ {((subringAlg β€˜β„fld)β€˜β„)} ∈ V)
2421, 23xpexd 7734 . . . . 5 (𝐼 ∈ 𝑉 β†’ (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) ∈ V)
2518, 20, 24prdssca 17398 . . . 4 (𝐼 ∈ 𝑉 β†’ ℝfld = (Scalarβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))))
26 fvex 6901 . . . . 5 (Baseβ€˜π») ∈ V
27 eqid 2732 . . . . . 6 ((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)) = ((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))
28 eqid 2732 . . . . . 6 (Scalarβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (Scalarβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})))
2927, 28resssca 17284 . . . . 5 ((Baseβ€˜π») ∈ V β†’ (Scalarβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (Scalarβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))))
3026, 29mp1i 13 . . . 4 (𝐼 ∈ 𝑉 β†’ (Scalarβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (Scalarβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))))
3125, 30eqtrd 2772 . . 3 (𝐼 ∈ 𝑉 β†’ ℝfld = (Scalarβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»))))
3211, 17, 313eqtr4d 2782 . 2 (𝐼 ∈ 𝑉 β†’ (Scalarβ€˜(toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs (Baseβ€˜π»)))) = ℝfld)
334, 32eqtrd 2772 1 (𝐼 ∈ 𝑉 β†’ (Scalarβ€˜π») = ℝfld)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4627   ↦ cmpt 5230   Γ— cxp 5673  β€˜cfv 6540  (class class class)co 7405  β„cr 11105  βˆšcsqrt 15176  Basecbs 17140   β†Ύs cress 17169  Scalarcsca 17196  Β·π‘–cip 17198  Xscprds 17387  Fieldcfield 20308  subringAlg csra 20773  β„fldcrefld 21148   toNrmGrp ctng 24078  toβ„‚PreHilctcph 24675  β„^crrx 24891
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185  ax-mulf 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-rp 12971  df-fz 13481  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-starv 17208  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-unif 17216  df-hom 17217  df-cco 17218  df-0g 17383  df-prds 17389  df-pws 17391  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-subg 18997  df-cmn 19644  df-mgp 19982  df-ur 19999  df-ring 20051  df-cring 20052  df-oppr 20142  df-dvdsr 20163  df-unit 20164  df-invr 20194  df-dvr 20207  df-drng 20309  df-field 20310  df-subrg 20353  df-sra 20777  df-rgmod 20778  df-cnfld 20937  df-refld 21149  df-dsmm 21278  df-frlm 21293  df-tng 24084  df-tcph 24677  df-rrx 24893
This theorem is referenced by:  rrxlines  47372
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