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Theorem rrxip 25238
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 25237 . . 3 (𝐼 ∈ 𝑉 β†’ 𝐻 = (toβ„‚PreHilβ€˜((ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) β†Ύs 𝐡)))
43fveq2d 6895 . 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 2731 . . . 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 2731 . . . 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 25073 . . 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 6905 . . . . 5 𝐡 ∈ V
9 eqid 2731 . . . . . 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 2731 . . . . . 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 17297 . . . . 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 2731 . . . . . 6 (ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})) = (ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))
14 refld 21482 . . . . . . 7 ℝfld ∈ Field
1514a1i 11 . . . . . 6 (𝐼 ∈ 𝑉 β†’ ℝfld ∈ Field)
16 snex 5431 . . . . . . 7 {((subringAlg β€˜β„fld)β€˜β„)} ∈ V
17 xpexg 7741 . . . . . . 7 ((𝐼 ∈ 𝑉 ∧ {((subringAlg β€˜β„fld)β€˜β„)} ∈ V) β†’ (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) ∈ V)
1816, 17mpan2 688 . . . . . 6 (𝐼 ∈ 𝑉 β†’ (𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}) ∈ V)
19 eqid 2731 . . . . . 6 (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})))
20 fvex 6904 . . . . . . . . 9 ((subringAlg β€˜β„fld)β€˜β„) ∈ V
2120snnz 4780 . . . . . . . 8 {((subringAlg β€˜β„fld)β€˜β„)} β‰  βˆ…
22 dmxp 5928 . . . . . . . 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 17414 . . . . 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 17410 . . . . . . 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 2732 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ ((subringAlg β€˜β„fld)β€˜β„) = ((subringAlg β€˜β„fld)β€˜β„))
28 rebase 21469 . . . . . . . . . . . . 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 21026 . . . . . . . . . 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 7207 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ ((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯) = ((subringAlg β€˜β„fld)β€˜β„))
3433fveq2d 6895 . . . . . . . . . 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 2782 . . . . . . . . 9 (π‘₯ ∈ 𝐼 β†’ (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = ℝ)
3635adantl 481 . . . . . . . 8 ((𝐼 ∈ 𝑉 ∧ π‘₯ ∈ 𝐼) β†’ (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = ℝ)
3736ixpeq2dva 8912 . . . . . . 7 (𝐼 ∈ 𝑉 β†’ Xπ‘₯ ∈ 𝐼 (Baseβ€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = Xπ‘₯ ∈ 𝐼 ℝ)
38 reex 11207 . . . . . . . 8 ℝ ∈ V
39 ixpconstg 8906 . . . . . . . 8 ((𝐼 ∈ 𝑉 ∧ ℝ ∈ V) β†’ Xπ‘₯ ∈ 𝐼 ℝ = (ℝ ↑m 𝐼))
4038, 39mpan2 688 . . . . . . 7 (𝐼 ∈ 𝑉 β†’ Xπ‘₯ ∈ 𝐼 ℝ = (ℝ ↑m 𝐼))
4126, 37, 403eqtrd 2775 . . . . . 6 (𝐼 ∈ 𝑉 β†’ (Baseβ€˜(ℝfldXs(𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)}))) = (ℝ ↑m 𝐼))
42 remulr 21474 . . . . . . . . . . 11 Β· = (.rβ€˜β„fld)
4333, 30sraip 21036 . . . . . . . . . . 11 (π‘₯ ∈ 𝐼 β†’ (.rβ€˜β„fld) = (Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)))
4442, 43eqtr2id 2784 . . . . . . . . . 10 (π‘₯ ∈ 𝐼 β†’ (Β·π‘–β€˜((𝐼 Γ— {((subringAlg β€˜β„fld)β€˜β„)})β€˜π‘₯)) = Β· )
4544oveqd 7429 . . . . . . . . 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 7428 . . . . . 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 7487 . . . . 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 2771 . . . 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 2785 . . 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 2785 . 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 2772 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 1540   ∈ wcel 2105   β‰  wne 2939  Vcvv 3473   βŠ† wss 3948  βˆ…c0 4322  {csn 4628   ↦ cmpt 5231   Γ— cxp 5674  dom cdm 5676  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414   ↑m cmap 8826  Xcixp 8897  β„cr 11115   Β· cmul 11121  Basecbs 17151   β†Ύs cress 17180  .rcmulr 17205  Β·π‘–cip 17209   Ξ£g cgsu 17393  Xscprds 17398  Fieldcfield 20584  subringAlg csra 21015  β„fldcrefld 21467  toβ„‚PreHilctcph 25015  β„^crrx 25231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-addf 11195  ax-mulf 11196
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-tpos 8217  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-rp 12982  df-fz 13492  df-seq 13974  df-exp 14035  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-starv 17219  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-unif 17227  df-hom 17228  df-cco 17229  df-0g 17394  df-prds 17400  df-pws 17402  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-subg 19046  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-ring 20136  df-cring 20137  df-oppr 20232  df-dvdsr 20255  df-unit 20256  df-invr 20286  df-dvr 20299  df-subrng 20442  df-subrg 20467  df-drng 20585  df-field 20586  df-sra 21019  df-rgmod 21020  df-cnfld 21234  df-refld 21468  df-dsmm 21597  df-frlm 21612  df-tng 24413  df-tcph 25017  df-rrx 25233
This theorem is referenced by:  rrxnm  25239
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