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Theorem frlmip 20543
 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 2758 . . . . . . 7 (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
3 eqid 2758 . . . . . . 7 (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
42, 3frlmpws 20515 . . . . . 6 ((𝑅𝑉𝐼𝑊) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
54ancoms 462 . . . . 5 ((𝐼𝑊𝑅𝑉) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
6 frlmphl.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
76ressid 16617 . . . . . . . . . 10 (𝑅𝑉 → (𝑅s 𝐵) = 𝑅)
8 eqidd 2759 . . . . . . . . . . 11 (𝑅𝑉 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
96eqimssi 3950 . . . . . . . . . . . 12 𝐵 ⊆ (Base‘𝑅)
109a1i 11 . . . . . . . . . . 11 (𝑅𝑉𝐵 ⊆ (Base‘𝑅))
118, 10srasca 20021 . . . . . . . . . 10 (𝑅𝑉 → (𝑅s 𝐵) = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
127, 11eqtr3d 2795 . . . . . . . . 9 (𝑅𝑉𝑅 = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
1312oveq1d 7165 . . . . . . . 8 (𝑅𝑉 → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
1413adantl 485 . . . . . . 7 ((𝐼𝑊𝑅𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
15 fvex 6671 . . . . . . . . 9 ((subringAlg ‘𝑅)‘𝐵) ∈ V
16 rlmval 20031 . . . . . . . . . . . 12 (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
176fveq2i 6661 . . . . . . . . . . . 12 ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
1816, 17eqtr4i 2784 . . . . . . . . . . 11 (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘𝐵)
1918oveq1i 7160 . . . . . . . . . 10 ((ringLMod‘𝑅) ↑s 𝐼) = (((subringAlg ‘𝑅)‘𝐵) ↑s 𝐼)
20 eqid 2758 . . . . . . . . . 10 (Scalar‘((subringAlg ‘𝑅)‘𝐵)) = (Scalar‘((subringAlg ‘𝑅)‘𝐵))
2119, 20pwsval 16817 . . . . . . . . 9 ((((subringAlg ‘𝑅)‘𝐵) ∈ V ∧ 𝐼𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
2215, 21mpan 689 . . . . . . . 8 (𝐼𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
2322adantr 484 . . . . . . 7 ((𝐼𝑊𝑅𝑉) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
2414, 23eqtr4d 2796 . . . . . 6 ((𝐼𝑊𝑅𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((ringLMod‘𝑅) ↑s 𝐼))
251fveq2i 6661 . . . . . . 7 (Base‘𝑌) = (Base‘(𝑅 freeLMod 𝐼))
2625a1i 11 . . . . . 6 ((𝐼𝑊𝑅𝑉) → (Base‘𝑌) = (Base‘(𝑅 freeLMod 𝐼)))
2724, 26oveq12d 7168 . . . . 5 ((𝐼𝑊𝑅𝑉) → ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
285, 27eqtr4d 2796 . . . 4 ((𝐼𝑊𝑅𝑉) → (𝑅 freeLMod 𝐼) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
291, 28syl5eq 2805 . . 3 ((𝐼𝑊𝑅𝑉) → 𝑌 = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
3029fveq2d 6662 . 2 ((𝐼𝑊𝑅𝑉) → (·𝑖𝑌) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
31 fvex 6671 . . . 4 (Base‘𝑌) ∈ V
32 eqid 2758 . . . . 5 ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))
33 eqid 2758 . . . . 5 (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
3432, 33ressip 16710 . . . 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 2758 . . . . 5 (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))
37 simpr 488 . . . . 5 ((𝐼𝑊𝑅𝑉) → 𝑅𝑉)
38 snex 5300 . . . . . . 7 {((subringAlg ‘𝑅)‘𝐵)} ∈ V
39 xpexg 7471 . . . . . . 7 ((𝐼𝑊 ∧ {((subringAlg ‘𝑅)‘𝐵)} ∈ V) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
4038, 39mpan2 690 . . . . . 6 (𝐼𝑊 → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
4140adantr 484 . . . . 5 ((𝐼𝑊𝑅𝑉) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
42 eqid 2758 . . . . 5 (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
4315snnz 4669 . . . . . 6 {((subringAlg ‘𝑅)‘𝐵)} ≠ ∅
44 dmxp 5770 . . . . . 6 ({((subringAlg ‘𝑅)‘𝐵)} ≠ ∅ → dom (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) = 𝐼)
4543, 44mp1i 13 . . . . 5 ((𝐼𝑊𝑅𝑉) → dom (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) = 𝐼)
4636, 37, 41, 42, 45, 33prdsip 16792 . . . 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))))))
4736, 37, 41, 42, 45prdsbas 16788 . . . . . 6 ((𝐼𝑊𝑅𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = X𝑥𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
48 eqidd 2759 . . . . . . . . . 10 (𝑥𝐼 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
499a1i 11 . . . . . . . . . 10 (𝑥𝐼𝐵 ⊆ (Base‘𝑅))
5048, 49srabase 20018 . . . . . . . . 9 (𝑥𝐼 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
516a1i 11 . . . . . . . . 9 (𝑥𝐼𝐵 = (Base‘𝑅))
5215fvconst2 6957 . . . . . . . . . 10 (𝑥𝐼 → ((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥) = ((subringAlg ‘𝑅)‘𝐵))
5352fveq2d 6662 . . . . . . . . 9 (𝑥𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
5450, 51, 533eqtr4rd 2804 . . . . . . . 8 (𝑥𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
5554adantl 485 . . . . . . 7 (((𝐼𝑊𝑅𝑉) ∧ 𝑥𝐼) → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
5655ixpeq2dva 8494 . . . . . 6 ((𝐼𝑊𝑅𝑉) → X𝑥𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = X𝑥𝐼 𝐵)
576fvexi 6672 . . . . . . . 8 𝐵 ∈ V
58 ixpconstg 8488 . . . . . . . 8 ((𝐼𝑊𝐵 ∈ V) → X𝑥𝐼 𝐵 = (𝐵m 𝐼))
5957, 58mpan2 690 . . . . . . 7 (𝐼𝑊X𝑥𝐼 𝐵 = (𝐵m 𝐼))
6059adantr 484 . . . . . 6 ((𝐼𝑊𝑅𝑉) → X𝑥𝐼 𝐵 = (𝐵m 𝐼))
6147, 56, 603eqtrd 2797 . . . . 5 ((𝐼𝑊𝑅𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝐵m 𝐼))
62 frlmphl.t . . . . . . . . . 10 · = (.r𝑅)
6352, 49sraip 20023 . . . . . . . . . 10 (𝑥𝐼 → (.r𝑅) = (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
6462, 63syl5req 2806 . . . . . . . . 9 (𝑥𝐼 → (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = · )
6564oveqd 7167 . . . . . . . 8 (𝑥𝐼 → ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥)) = ((𝑓𝑥) · (𝑔𝑥)))
6665mpteq2ia 5123 . . . . . . 7 (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥)))
6766oveq2i 7161 . . . . . 6 (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥)))) = (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))
6867a1i 11 . . . . 5 ((𝐼𝑊𝑅𝑉) → (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥)))) = (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥)))))
6961, 61, 68mpoeq123dv 7223 . . . 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))))))
7046, 69eqtrd 2793 . . 3 ((𝐼𝑊𝑅𝑉) → (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝑓 ∈ (𝐵m 𝐼), 𝑔 ∈ (𝐵m 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))))
7135, 70syl5eqr 2807 . 2 ((𝐼𝑊𝑅𝑉) → (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))) = (𝑓 ∈ (𝐵m 𝐼), 𝑔 ∈ (𝐵m 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))))
7230, 71eqtr2d 2794 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 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  {csn 4522   ↦ cmpt 5112   × cxp 5522  dom cdm 5524  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑m cmap 8416  Xcixp 8479  Basecbs 16541   ↾s cress 16542  .rcmulr 16624  Scalarcsca 16626  ·𝑖cip 16628   Σg cgsu 16772  Xscprds 16777   ↑s cpws 16778  subringAlg csra 20008  ringLModcrglmod 20009   freeLMod cfrlm 20511 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-ixp 8480  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-5 11740  df-6 11741  df-7 11742  df-8 11743  df-9 11744  df-n0 11935  df-z 12021  df-dec 12138  df-uz 12283  df-fz 12940  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-mulr 16637  df-sca 16639  df-vsca 16640  df-ip 16641  df-tset 16642  df-ple 16643  df-ds 16645  df-hom 16647  df-cco 16648  df-prds 16779  df-pws 16781  df-sra 20012  df-rgmod 20013  df-dsmm 20497  df-frlm 20512 This theorem is referenced by:  frlmipval  20544  frlmphl  20546
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