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Theorem frlmip 21184
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 2736 . . . . . . 7 (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
3 eqid 2736 . . . . . . 7 (Base‘(𝑅 freeLMod 𝐼)) = (Base‘(𝑅 freeLMod 𝐼))
42, 3frlmpws 21156 . . . . . 6 ((𝑅𝑉𝐼𝑊) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
54ancoms 459 . . . . 5 ((𝐼𝑊𝑅𝑉) → (𝑅 freeLMod 𝐼) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
6 frlmphl.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
76ressid 17125 . . . . . . . . . 10 (𝑅𝑉 → (𝑅s 𝐵) = 𝑅)
8 eqidd 2737 . . . . . . . . . . 11 (𝑅𝑉 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
96eqimssi 4002 . . . . . . . . . . . 12 𝐵 ⊆ (Base‘𝑅)
109a1i 11 . . . . . . . . . . 11 (𝑅𝑉𝐵 ⊆ (Base‘𝑅))
118, 10srasca 20646 . . . . . . . . . 10 (𝑅𝑉 → (𝑅s 𝐵) = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
127, 11eqtr3d 2778 . . . . . . . . 9 (𝑅𝑉𝑅 = (Scalar‘((subringAlg ‘𝑅)‘𝐵)))
1312oveq1d 7372 . . . . . . . 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 6855 . . . . . . . . 9 ((subringAlg ‘𝑅)‘𝐵) ∈ V
16 rlmval 20660 . . . . . . . . . . . 12 (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
176fveq2i 6845 . . . . . . . . . . . 12 ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘(Base‘𝑅))
1816, 17eqtr4i 2767 . . . . . . . . . . 11 (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘𝐵)
1918oveq1i 7367 . . . . . . . . . 10 ((ringLMod‘𝑅) ↑s 𝐼) = (((subringAlg ‘𝑅)‘𝐵) ↑s 𝐼)
20 eqid 2736 . . . . . . . . . 10 (Scalar‘((subringAlg ‘𝑅)‘𝐵)) = (Scalar‘((subringAlg ‘𝑅)‘𝐵))
2119, 20pwsval 17368 . . . . . . . . 9 ((((subringAlg ‘𝑅)‘𝐵) ∈ V ∧ 𝐼𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘((subringAlg ‘𝑅)‘𝐵))Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
2215, 21mpan 688 . . . . . . . 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 2779 . . . . . 6 ((𝐼𝑊𝑅𝑉) → (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = ((ringLMod‘𝑅) ↑s 𝐼))
251fveq2i 6845 . . . . . . 7 (Base‘𝑌) = (Base‘(𝑅 freeLMod 𝐼))
2625a1i 11 . . . . . 6 ((𝐼𝑊𝑅𝑉) → (Base‘𝑌) = (Base‘(𝑅 freeLMod 𝐼)))
2724, 26oveq12d 7375 . . . . 5 ((𝐼𝑊𝑅𝑉) → ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘(𝑅 freeLMod 𝐼))))
285, 27eqtr4d 2779 . . . 4 ((𝐼𝑊𝑅𝑉) → (𝑅 freeLMod 𝐼) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
291, 28eqtrid 2788 . . 3 ((𝐼𝑊𝑅𝑉) → 𝑌 = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)))
3029fveq2d 6846 . 2 ((𝐼𝑊𝑅𝑉) → (·𝑖𝑌) = (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))))
31 fvex 6855 . . . 4 (Base‘𝑌) ∈ V
32 eqid 2736 . . . . 5 ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌)) = ((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))
33 eqid 2736 . . . . 5 (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
3432, 33ressip 17226 . . . 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 2736 . . . . 5 (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) = (𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))
37 simpr 485 . . . . 5 ((𝐼𝑊𝑅𝑉) → 𝑅𝑉)
38 snex 5388 . . . . . . 7 {((subringAlg ‘𝑅)‘𝐵)} ∈ V
39 xpexg 7684 . . . . . . 7 ((𝐼𝑊 ∧ {((subringAlg ‘𝑅)‘𝐵)} ∈ V) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
4038, 39mpan2 689 . . . . . 6 (𝐼𝑊 → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
4140adantr 481 . . . . 5 ((𝐼𝑊𝑅𝑉) → (𝐼 × {((subringAlg ‘𝑅)‘𝐵)}) ∈ V)
42 eqid 2736 . . . . 5 (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})))
4315snnz 4737 . . . . . 6 {((subringAlg ‘𝑅)‘𝐵)} ≠ ∅
44 dmxp 5884 . . . . . 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 17343 . . . 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 17339 . . . . . 6 ((𝐼𝑊𝑅𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = X𝑥𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
48 eqidd 2737 . . . . . . . . . 10 (𝑥𝐼 → ((subringAlg ‘𝑅)‘𝐵) = ((subringAlg ‘𝑅)‘𝐵))
499a1i 11 . . . . . . . . . 10 (𝑥𝐼𝐵 ⊆ (Base‘𝑅))
5048, 49srabase 20640 . . . . . . . . 9 (𝑥𝐼 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
516a1i 11 . . . . . . . . 9 (𝑥𝐼𝐵 = (Base‘𝑅))
5215fvconst2 7153 . . . . . . . . . 10 (𝑥𝐼 → ((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥) = ((subringAlg ‘𝑅)‘𝐵))
5352fveq2d 6846 . . . . . . . . 9 (𝑥𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = (Base‘((subringAlg ‘𝑅)‘𝐵)))
5450, 51, 533eqtr4rd 2787 . . . . . . . 8 (𝑥𝐼 → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
5554adantl 482 . . . . . . 7 (((𝐼𝑊𝑅𝑉) ∧ 𝑥𝐼) → (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = 𝐵)
5655ixpeq2dva 8850 . . . . . 6 ((𝐼𝑊𝑅𝑉) → X𝑥𝐼 (Base‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = X𝑥𝐼 𝐵)
576fvexi 6856 . . . . . . . 8 𝐵 ∈ V
58 ixpconstg 8844 . . . . . . . 8 ((𝐼𝑊𝐵 ∈ V) → X𝑥𝐼 𝐵 = (𝐵m 𝐼))
5957, 58mpan2 689 . . . . . . 7 (𝐼𝑊X𝑥𝐼 𝐵 = (𝐵m 𝐼))
6059adantr 481 . . . . . 6 ((𝐼𝑊𝑅𝑉) → X𝑥𝐼 𝐵 = (𝐵m 𝐼))
6147, 56, 603eqtrd 2780 . . . . 5 ((𝐼𝑊𝑅𝑉) → (Base‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝐵m 𝐼))
62 frlmphl.t . . . . . . . . . 10 · = (.r𝑅)
6352, 49sraip 20650 . . . . . . . . . 10 (𝑥𝐼 → (.r𝑅) = (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)))
6462, 63eqtr2id 2789 . . . . . . . . 9 (𝑥𝐼 → (·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥)) = · )
6564oveqd 7374 . . . . . . . 8 (𝑥𝐼 → ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥)) = ((𝑓𝑥) · (𝑔𝑥)))
6665mpteq2ia 5208 . . . . . . 7 (𝑥𝐼 ↦ ((𝑓𝑥)(·𝑖‘((𝐼 × {((subringAlg ‘𝑅)‘𝐵)})‘𝑥))(𝑔𝑥))) = (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥)))
6766oveq2i 7368 . . . . . 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 7432 . . . 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 2776 . . 3 ((𝐼𝑊𝑅𝑉) → (·𝑖‘(𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)}))) = (𝑓 ∈ (𝐵m 𝐼), 𝑔 ∈ (𝐵m 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))))
7135, 70eqtr3id 2790 . 2 ((𝐼𝑊𝑅𝑉) → (·𝑖‘((𝑅Xs(𝐼 × {((subringAlg ‘𝑅)‘𝐵)})) ↾s (Base‘𝑌))) = (𝑓 ∈ (𝐵m 𝐼), 𝑔 ∈ (𝐵m 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))))
7230, 71eqtr2d 2777 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 396   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  wss 3910  c0 4282  {csn 4586  cmpt 5188   × cxp 5631  dom cdm 5633  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  Xcixp 8835  Basecbs 17083  s cress 17112  .rcmulr 17134  Scalarcsca 17136  ·𝑖cip 17138   Σg cgsu 17322  Xscprds 17327  s cpws 17328  subringAlg csra 20629  ringLModcrglmod 20630   freeLMod cfrlm 21152
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 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-prds 17329  df-pws 17331  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153
This theorem is referenced by:  frlmipval  21185  frlmphl  21187
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