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Theorem frlmip 20036
Description: The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)
Hypotheses
Ref Expression
frlmphl.y 𝑌 = (𝑅 freeLMod 𝐼)
frlmphl.b 𝐵 = (Base‘𝑅)
frlmphl.t · = (.r𝑅)
Assertion
Ref Expression
frlmip ((𝐼𝑊𝑅𝑉) → (𝑓 ∈ (𝐵𝑚 𝐼), 𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))) = (·𝑖𝑌))
Distinct variable groups:   𝐵,𝑓,𝑔,𝑥   𝑓,𝐼,𝑔,𝑥   𝑅,𝑓,𝑔,𝑥   𝑓,𝑉,𝑔,𝑥   𝑓,𝑊,𝑔,𝑥
Allowed substitution hints:   · (𝑥,𝑓,𝑔)   𝑌(𝑥,𝑓,𝑔)

Proof of Theorem frlmip
StepHypRef Expression
1 frlmphl.y . . . 4 𝑌 = (𝑅 freeLMod 𝐼)
2 eqid 2621 . . . . . . 7 (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
3 eqid 2621 . . . . . . 7 (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
42, 3frlmpws 20013 . . . . . 6 ((𝑅𝑉𝐼𝑊) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
54ancoms 469 . . . . 5 ((𝐼𝑊𝑅𝑉) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
6 frlmphl.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
76ressid 15856 . . . . . . . . . 10 (𝑅𝑉 → (𝑅s 𝐵) = 𝑅)
8 eqidd 2622 . . . . . . . . . . 11 (𝑅𝑉 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
96eqimssi 3638 . . . . . . . . . . . 12 𝐵 ⊆ (Base‘𝑅)
109a1i 11 . . . . . . . . . . 11 (𝑅𝑉𝐵 ⊆ (Base‘𝑅))
118, 10srasca 19100 . . . . . . . . . 10 (𝑅𝑉 → (𝑅s 𝐵) = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
127, 11eqtr3d 2657 . . . . . . . . 9 (𝑅𝑉𝑅 = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
1312oveq1d 6619 . . . . . . . 8 (𝑅𝑉 → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
1413adantl 482 . . . . . . 7 ((𝐼𝑊𝑅𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
15 fvex 6158 . . . . . . . . 9 ((subringAlg ‘𝑅)‘𝐵) ∈ V
16 rlmval 19110 . . . . . . . . . . . 12 (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
176fveq2i 6151 . . . . . . . . . . . 12 ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
1816, 17eqtr4i 2646 . . . . . . . . . . 11 (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘𝐵)
1918oveq1i 6614 . . . . . . . . . 10 ((ringLMod‘𝑅) ↑s 𝐼) = (((subringAlg ‘𝑅)‘𝐵) ↑s 𝐼)
20 eqid 2621 . . . . . . . . . 10 (Scalar‘((subringAlg ‘𝑅)‘𝐵)) = (Scalar‘((subringAlg ‘𝑅)‘𝐵))
2119, 20pwsval 16067 . . . . . . . . 9 ((((subringAlg ‘𝑅)‘𝐵) ∈ V ∧ 𝐼𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
2215, 21mpan 705 . . . . . . . 8 (𝐼𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
2322adantr 481 . . . . . . 7 ((𝐼𝑊𝑅𝑉) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
2414, 23eqtr4d 2658 . . . . . 6 ((𝐼𝑊𝑅𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((ringLMod‘𝑅) ↑s 𝐼))
251fveq2i 6151 . . . . . . 7 (Base‘𝑌) = (Base‘(𝑅 freeLMod 𝐼))
2625a1i 11 . . . . . 6 ((𝐼𝑊𝑅𝑉) → (Base‘𝑌) = (Base‘(𝑅 freeLMod 𝐼)))
2724, 26oveq12d 6622 . . . . 5 ((𝐼𝑊𝑅𝑉) → ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
285, 27eqtr4d 2658 . . . 4 ((𝐼𝑊𝑅𝑉) → (𝑅 freeLMod 𝐼) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
291, 28syl5eq 2667 . . 3 ((𝐼𝑊𝑅𝑉) → 𝑌 = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
3029fveq2d 6152 . 2 ((𝐼𝑊𝑅𝑉) → (·𝑖𝑌) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
31 fvex 6158 . . . 4 (Base‘𝑌) ∈ V
32 eqid 2621 . . . . 5 ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))
33 eqid 2621 . . . . 5 (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
3432, 33ressip 15954 . . . 4 ((Base‘𝑌) ∈ V → (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
3531, 34ax-mp 5 . . 3 (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
36 eqid 2621 . . . . 5 (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))
37 simpr 477 . . . . 5 ((𝐼𝑊𝑅𝑉) → 𝑅𝑉)
38 snex 4869 . . . . . . 7 {((subringAlg ‘𝑅)‘𝐵)} ∈ V
39 xpexg 6913 . . . . . . 7 ((𝐼𝑊 ∧ {((subringAlg ‘𝑅)‘𝐵)} ∈ V) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
4038, 39mpan2 706 . . . . . 6 (𝐼𝑊 → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
4140adantr 481 . . . . 5 ((𝐼𝑊𝑅𝑉) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
42 eqid 2621 . . . . 5 (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
4315snnz 4279 . . . . . . 7 {((subringAlg ‘𝑅)‘𝐵)} ≠ ∅
44 dmxp 5304 . . . . . . 7 ({((subringAlg ‘𝑅)‘𝐵)} ≠ ∅ → dom (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) = 𝐼)
4543, 44ax-mp 5 . . . . . 6 dom (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) = 𝐼
4645a1i 11 . . . . 5 ((𝐼𝑊𝑅𝑉) → dom (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) = 𝐼)
4736, 37, 41, 42, 46, 33prdsip 16042 . . . 4 ((𝐼𝑊𝑅𝑉) → (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝑓 ∈ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))), 𝑔 ∈ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥))))))
4836, 37, 41, 42, 46prdsbas 16038 . . . . . 6 ((𝐼𝑊𝑅𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = X𝑥𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
49 eqidd 2622 . . . . . . . . . 10 (𝑥𝐼 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
509a1i 11 . . . . . . . . . 10 (𝑥𝐼𝐵 ⊆ (Base‘𝑅))
5149, 50srabase 19097 . . . . . . . . 9 (𝑥𝐼 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
526a1i 11 . . . . . . . . 9 (𝑥𝐼𝐵 = (Base‘𝑅))
5315fvconst2 6423 . . . . . . . . . 10 (𝑥𝐼 → ((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥) = ((subringAlg ‘𝑅)‘𝐵))
5453fveq2d 6152 . . . . . . . . 9 (𝑥𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
5551, 52, 543eqtr4rd 2666 . . . . . . . 8 (𝑥𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
5655adantl 482 . . . . . . 7 (((𝐼𝑊𝑅𝑉) ∧ 𝑥𝐼) → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
5756ixpeq2dva 7867 . . . . . 6 ((𝐼𝑊𝑅𝑉) → X𝑥𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = X𝑥𝐼 𝐵)
58 fvex 6158 . . . . . . . . 9 (Base‘𝑅) ∈ V
596, 58eqeltri 2694 . . . . . . . 8 𝐵 ∈ V
60 ixpconstg 7861 . . . . . . . 8 ((𝐼𝑊𝐵 ∈ V) → X𝑥𝐼 𝐵 = (𝐵𝑚 𝐼))
6159, 60mpan2 706 . . . . . . 7 (𝐼𝑊X𝑥𝐼 𝐵 = (𝐵𝑚 𝐼))
6261adantr 481 . . . . . 6 ((𝐼𝑊𝑅𝑉) → X𝑥𝐼 𝐵 = (𝐵𝑚 𝐼))
6348, 57, 623eqtrd 2659 . . . . 5 ((𝐼𝑊𝑅𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝐵𝑚 𝐼))
64 frlmphl.t . . . . . . . . . 10 · = (.r𝑅)
6553, 50sraip 19102 . . . . . . . . . 10 (𝑥𝐼 → (.r𝑅) = (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
6664, 65syl5req 2668 . . . . . . . . 9 (𝑥𝐼 → (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = · )
6766oveqd 6621 . . . . . . . 8 (𝑥𝐼 → ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥)) = ((𝑓𝑥) · (𝑔𝑥)))
6867mpteq2ia 4700 . . . . . . 7 (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥)))
6968oveq2i 6615 . . . . . 6 (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥)))) = (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))
7069a1i 11 . . . . 5 ((𝐼𝑊𝑅𝑉) → (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥)))) = (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥)))))
7163, 63, 70mpt2eq123dv 6670 . . . 4 ((𝐼𝑊𝑅𝑉) → (𝑓 ∈ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))), 𝑔 ∈ (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥))))) = (𝑓 ∈ (𝐵𝑚 𝐼), 𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))))
7247, 71eqtrd 2655 . . 3 ((𝐼𝑊𝑅𝑉) → (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝑓 ∈ (𝐵𝑚 𝐼), 𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))))
7335, 72syl5eqr 2669 . 2 ((𝐼𝑊𝑅𝑉) → (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))) = (𝑓 ∈ (𝐵𝑚 𝐼), 𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))))
7430, 73eqtr2d 2656 1 ((𝐼𝑊𝑅𝑉) → (𝑓 ∈ (𝐵𝑚 𝐼), 𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))) = (·𝑖𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  wss 3555  c0 3891  {csn 4148  cmpt 4673   × cxp 5072  dom cdm 5074  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802  Xcixp 7852  Basecbs 15781  s cress 15782  .rcmulr 15863  Scalarcsca 15865  ·𝑖cip 15867   Σg cgsu 16022  Xscprds 16027  s cpws 16028  subringAlg csra 19087  ringLModcrglmod 19088   freeLMod cfrlm 20009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-prds 16029  df-pws 16031  df-sra 19091  df-rgmod 19092  df-dsmm 19995  df-frlm 20010
This theorem is referenced by:  frlmipval  20037  frlmphl  20039
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