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Theorem rrxip 24899
Description: The inner product of the generalized real Euclidean spaces. (Contributed by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
rrxval.r 𝐻 = (ℝ^β€˜πΌ)
rrxbase.b 𝐡 = (Baseβ€˜π»)
Assertion
Ref Expression
rrxip (𝐼 ∈ 𝑉 β†’ (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯))))) = (Β·π‘–β€˜π»))
Distinct variable groups:   𝑓,𝑔,π‘₯,𝐡   𝑓,𝐼,𝑔,π‘₯   𝑓,𝑉,𝑔,π‘₯
Allowed substitution hints:   𝐻(π‘₯,𝑓,𝑔)
Proof of Theorem rrxip
StepHypRef Expression
1 rrxval.r . . . 4 𝐻 = (ℝ^β€˜πΌ)
2 rrxbase.b . . . 4 𝐡 = (Baseβ€˜π»)
31, 2rrxprds 24898 . . 3 (𝐼 ∈ 𝑉 β†’ 𝐻 = (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)))
43fveq2d 6893 . 2 (𝐼 ∈ 𝑉 β†’ (Β·π‘–β€˜π») = (Β·π‘–β€˜(toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡))))
5 eqid 2733 . . . 4 (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)) = (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡))
6 eqid 2733 . . . 4 (Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)) = (Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡))
75, 6tcphip 24734 . . 3 (Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)) = (Β·π‘–β€˜(toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)))
82fvexi 6903 . . . . 5 𝐡 ∈ V
9 eqid 2733 . . . . . 6 ((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡) = ((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)
10 eqid 2733 . . . . . 6 (Β·π‘–β€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (Β·π‘–β€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})))
119, 10ressip 17287 . . . . 5 (𝐡 ∈ V β†’ (Β·π‘–β€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)))
128, 11ax-mp 5 . . . 4 (Β·π‘–β€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡))
13 eqid 2733 . . . . . 6 (ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) = (ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))
14 refld 21164 . . . . . . 7 ℝfld ∈ Field
1514a1i 11 . . . . . 6 (𝐼 ∈ 𝑉 β†’ ℝfld ∈ Field)
16 snex 5431 . . . . . . 7 {((subringAlg β€˜β„fld)β€˜β„)} ∈ V
17 xpexg 7734 . . . . . . 7 ((𝐼 ∈ 𝑉 ∧ {((subringAlg β€˜β„fld)β€˜β„)} ∈ V) β†’ (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) ∈ V)
1816, 17mpan2 690 . . . . . 6 (𝐼 ∈ 𝑉 β†’ (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) ∈ V)
19 eqid 2733 . . . . . 6 (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})))
20 fvex 6902 . . . . . . . . 9 ((subringAlg β€˜β„fld)β€˜β„) ∈ V
2120snnz 4780 . . . . . . . 8 {((subringAlg β€˜β„fld)β€˜β„)} β‰  βˆ…
22 dmxp 5927 . . . . . . . 8 ({((subringAlg β€˜β„fld)β€˜β„)} β‰  βˆ… β†’ dom (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) = 𝐼)
2321, 22ax-mp 5 . . . . . . 7 dom (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) = 𝐼
2423a1i 11 . . . . . 6 (𝐼 ∈ 𝑉 β†’ dom (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) = 𝐼)
2513, 15, 18, 19, 24, 10prdsip 17404 . . . . 5 (𝐼 ∈ 𝑉 β†’ (Β·π‘–β€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (𝑓 ∈ (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))), 𝑔 ∈ (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) ↦ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯))(π‘”β€˜π‘₯))))))
2613, 15, 18, 19, 24prdsbas 17400 . . . . . . 7 (𝐼 ∈ 𝑉 β†’ (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = Xπ‘₯ ∈ 𝐼 (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)))
27 eqidd 2734 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ ((subringAlg β€˜β„fld)β€˜β„) = ((subringAlg β€˜β„fld)β€˜β„))
28 rebase 21151 . . . . . . . . . . . . 13 ℝ = (Baseβ€˜β„fld)
2928eqimssi 4042 . . . . . . . . . . . 12 ℝ βŠ† (Baseβ€˜β„fld)
3029a1i 11 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ ℝ βŠ† (Baseβ€˜β„fld))
3127, 30srabase 20785 . . . . . . . . . 10 (π‘₯ ∈ 𝐼 β†’ (Baseβ€˜β„fld) = (Baseβ€˜((subringAlg β€˜β„fld)β€˜β„)))
3228a1i 11 . . . . . . . . . 10 (π‘₯ ∈ 𝐼 β†’ ℝ = (Baseβ€˜β„fld))
3320fvconst2 7202 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ ((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯) = ((subringAlg β€˜β„fld)β€˜β„))
3433fveq2d 6893 . . . . . . . . . 10 (π‘₯ ∈ 𝐼 β†’ (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = (Baseβ€˜((subringAlg β€˜β„fld)β€˜β„)))
3531, 32, 343eqtr4rd 2784 . . . . . . . . 9 (π‘₯ ∈ 𝐼 β†’ (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = ℝ)
3635adantl 483 . . . . . . . 8 ((𝐼 ∈ 𝑉 ∧ π‘₯ ∈ 𝐼) β†’ (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = ℝ)
3736ixpeq2dva 8903 . . . . . . 7 (𝐼 ∈ 𝑉 β†’ Xπ‘₯ ∈ 𝐼 (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = Xπ‘₯ ∈ 𝐼 ℝ)
38 reex 11198 . . . . . . . 8 ℝ ∈ V
39 ixpconstg 8897 . . . . . . . 8 ((𝐼 ∈ 𝑉 ∧ ℝ ∈ V) β†’ Xπ‘₯ ∈ 𝐼 ℝ = (ℝ ↑m 𝐼))
4038, 39mpan2 690 . . . . . . 7 (𝐼 ∈ 𝑉 β†’ Xπ‘₯ ∈ 𝐼 ℝ = (ℝ ↑m 𝐼))
4126, 37, 403eqtrd 2777 . . . . . 6 (𝐼 ∈ 𝑉 β†’ (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (ℝ ↑m 𝐼))
42 remulr 21156 . . . . . . . . . . 11 Β· = (.rβ€˜β„fld)
4333, 30sraip 20795 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ (.rβ€˜β„fld) = (Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)))
4442, 43eqtr2id 2786 . . . . . . . . . 10 (π‘₯ ∈ 𝐼 β†’ (Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = Β· )
4544oveqd 7423 . . . . . . . . 9 (π‘₯ ∈ 𝐼 β†’ ((π‘“β€˜π‘₯)(Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯))(π‘”β€˜π‘₯)) = ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯)))
4645mpteq2ia 5251 . . . . . . . 8 (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯))(π‘”β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯)))
4746a1i 11 . . . . . . 7 (𝐼 ∈ 𝑉 β†’ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯))(π‘”β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯))))
4847oveq2d 7422 . . . . . 6 (𝐼 ∈ 𝑉 β†’ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯))(π‘”β€˜π‘₯)))) = (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯)))))
4941, 41, 48mpoeq123dv 7481 . . . . 5 (𝐼 ∈ 𝑉 β†’ (𝑓 ∈ (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))), 𝑔 ∈ (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) ↦ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯))(π‘”β€˜π‘₯))))) = (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯))))))
5025, 49eqtrd 2773 . . . 4 (𝐼 ∈ 𝑉 β†’ (Β·π‘–β€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯))))))
5112, 50eqtr3id 2787 . . 3 (𝐼 ∈ 𝑉 β†’ (Β·π‘–β€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)) = (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯))))))
527, 51eqtr3id 2787 . 2 (𝐼 ∈ 𝑉 β†’ (Β·π‘–β€˜(toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡))) = (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯))))))
534, 52eqtr2d 2774 1 (𝐼 ∈ 𝑉 β†’ (𝑓 ∈ (ℝ ↑m 𝐼), 𝑔 ∈ (ℝ ↑m 𝐼) ↦ (ℝfld Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯) Β· (π‘”β€˜π‘₯))))) = (Β·π‘–β€˜π»))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475   βŠ† wss 3948  βˆ…c0 4322  {csn 4628   ↦ cmpt 5231   Γ— cxp 5674  dom cdm 5676  β€˜cfv 6541  (class class class)co 7406   ∈ cmpo 7408   ↑m cmap 8817  Xcixp 8888  β„cr 11106   Β· cmul 11112  Basecbs 17141   β†Ύs cress 17170  .rcmulr 17195  Β·π‘–cip 17199   Ξ£g cgsu 17383  Xscprds 17388  Fieldcfield 20309  subringAlg csra 20774  β„fldcrefld 21149  toβ„‚PreHilctcph 24676  β„^crrx 24892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185  ax-addf 11186  ax-mulf 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-tpos 8208  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-rp 12972  df-fz 13482  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-mulr 17208  df-starv 17209  df-sca 17210  df-vsca 17211  df-ip 17212  df-tset 17213  df-ple 17214  df-ds 17216  df-unif 17217  df-hom 17218  df-cco 17219  df-0g 17384  df-prds 17390  df-pws 17392  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-grp 18819  df-minusg 18820  df-subg 18998  df-cmn 19645  df-mgp 19983  df-ur 20000  df-ring 20052  df-cring 20053  df-oppr 20143  df-dvdsr 20164  df-unit 20165  df-invr 20195  df-dvr 20208  df-drng 20310  df-field 20311  df-subrg 20354  df-sra 20778  df-rgmod 20779  df-cnfld 20938  df-refld 21150  df-dsmm 21279  df-frlm 21294  df-tng 24085  df-tcph 24678  df-rrx 24894
This theorem is referenced by:  rrxnm  24900
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