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Theorem rrxip 24738
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 24737 . . 3 (𝐼 ∈ 𝑉 β†’ 𝐻 = (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)))
43fveq2d 6843 . 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 2736 . . . 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 2736 . . . 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 24573 . . 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 6853 . . . . 5 𝐡 ∈ V
9 eqid 2736 . . . . . 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 2736 . . . . . 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 17218 . . . . 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 2736 . . . . . 6 (ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) = (ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))
14 refld 21008 . . . . . . 7 ℝfld ∈ Field
1514a1i 11 . . . . . 6 (𝐼 ∈ 𝑉 β†’ ℝfld ∈ Field)
16 snex 5386 . . . . . . 7 {((subringAlg β€˜β„fld)β€˜β„)} ∈ V
17 xpexg 7680 . . . . . . 7 ((𝐼 ∈ 𝑉 ∧ {((subringAlg β€˜β„fld)β€˜β„)} ∈ V) β†’ (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) ∈ V)
1816, 17mpan2 689 . . . . . 6 (𝐼 ∈ 𝑉 β†’ (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) ∈ V)
19 eqid 2736 . . . . . 6 (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})))
20 fvex 6852 . . . . . . . . 9 ((subringAlg β€˜β„fld)β€˜β„) ∈ V
2120snnz 4735 . . . . . . . 8 {((subringAlg β€˜β„fld)β€˜β„)} β‰  βˆ…
22 dmxp 5882 . . . . . . . 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 17335 . . . . 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 17331 . . . . . . 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 2737 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ ((subringAlg β€˜β„fld)β€˜β„) = ((subringAlg β€˜β„fld)β€˜β„))
28 rebase 20995 . . . . . . . . . . . . 13 ℝ = (Baseβ€˜β„fld)
2928eqimssi 4000 . . . . . . . . . . . 12 ℝ βŠ† (Baseβ€˜β„fld)
3029a1i 11 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ ℝ βŠ† (Baseβ€˜β„fld))
3127, 30srabase 20625 . . . . . . . . . 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 7149 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ ((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯) = ((subringAlg β€˜β„fld)β€˜β„))
3433fveq2d 6843 . . . . . . . . . 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 2787 . . . . . . . . 9 (π‘₯ ∈ 𝐼 β†’ (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = ℝ)
3635adantl 482 . . . . . . . 8 ((𝐼 ∈ 𝑉 ∧ π‘₯ ∈ 𝐼) β†’ (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = ℝ)
3736ixpeq2dva 8846 . . . . . . 7 (𝐼 ∈ 𝑉 β†’ Xπ‘₯ ∈ 𝐼 (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = Xπ‘₯ ∈ 𝐼 ℝ)
38 reex 11138 . . . . . . . 8 ℝ ∈ V
39 ixpconstg 8840 . . . . . . . 8 ((𝐼 ∈ 𝑉 ∧ ℝ ∈ V) β†’ Xπ‘₯ ∈ 𝐼 ℝ = (ℝ ↑m 𝐼))
4038, 39mpan2 689 . . . . . . 7 (𝐼 ∈ 𝑉 β†’ Xπ‘₯ ∈ 𝐼 ℝ = (ℝ ↑m 𝐼))
4126, 37, 403eqtrd 2780 . . . . . 6 (𝐼 ∈ 𝑉 β†’ (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (ℝ ↑m 𝐼))
42 remulr 21000 . . . . . . . . . . 11 Β· = (.rβ€˜β„fld)
4333, 30sraip 20635 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ (.rβ€˜β„fld) = (Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)))
4442, 43eqtr2id 2789 . . . . . . . . . 10 (π‘₯ ∈ 𝐼 β†’ (Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = Β· )
4544oveqd 7370 . . . . . . . . 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 5206 . . . . . . . 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 7369 . . . . . 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 7428 . . . . 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 2776 . . . 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 2790 . . 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 2790 . 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 2777 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 1541   ∈ wcel 2106   β‰  wne 2941  Vcvv 3443   βŠ† wss 3908  βˆ…c0 4280  {csn 4584   ↦ cmpt 5186   Γ— cxp 5629  dom cdm 5631  β€˜cfv 6493  (class class class)co 7353   ∈ cmpo 7355   ↑m cmap 8761  Xcixp 8831  β„cr 11046   Β· cmul 11052  Basecbs 17075   β†Ύs cress 17104  .rcmulr 17126  Β·π‘–cip 17130   Ξ£g cgsu 17314  Xscprds 17319  Fieldcfield 20171  subringAlg csra 20614  β„fldcrefld 20993  toβ„‚PreHilctcph 24515  β„^crrx 24731
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 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124  ax-pre-sup 11125  ax-addf 11126  ax-mulf 11127
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-tpos 8153  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-1o 8408  df-er 8644  df-map 8763  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9374  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-div 11809  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12410  df-z 12496  df-dec 12615  df-uz 12760  df-rp 12908  df-fz 13417  df-seq 13899  df-exp 13960  df-cj 14976  df-re 14977  df-im 14978  df-sqrt 15112  df-abs 15113  df-struct 17011  df-sets 17028  df-slot 17046  df-ndx 17058  df-base 17076  df-ress 17105  df-plusg 17138  df-mulr 17139  df-starv 17140  df-sca 17141  df-vsca 17142  df-ip 17143  df-tset 17144  df-ple 17145  df-ds 17147  df-unif 17148  df-hom 17149  df-cco 17150  df-0g 17315  df-prds 17321  df-pws 17323  df-mgm 18489  df-sgrp 18538  df-mnd 18549  df-grp 18743  df-minusg 18744  df-subg 18916  df-cmn 19555  df-mgp 19888  df-ur 19905  df-ring 19952  df-cring 19953  df-oppr 20034  df-dvdsr 20055  df-unit 20056  df-invr 20086  df-dvr 20097  df-drng 20172  df-field 20173  df-subrg 20205  df-sra 20618  df-rgmod 20619  df-cnfld 20782  df-refld 20994  df-dsmm 21123  df-frlm 21138  df-tng 23924  df-tcph 24517  df-rrx 24733
This theorem is referenced by:  rrxnm  24739
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