Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcoop Structured version   Visualization version   GIF version

Theorem lcoop 41462
Description: A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lcoop.b 𝐵 = (Base‘𝑀)
lcoop.s 𝑆 = (Scalar‘𝑀)
lcoop.r 𝑅 = (Base‘𝑆)
Assertion
Ref Expression
lcoop ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
Distinct variable groups:   𝐵,𝑐   𝑀,𝑐,𝑠   𝑅,𝑐,𝑠   𝑉,𝑐,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝑆(𝑠,𝑐)   𝑋(𝑠,𝑐)

Proof of Theorem lcoop
Dummy variables 𝑚 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3203 . . 3 (𝑀𝑋𝑀 ∈ V)
21adantr 481 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → 𝑀 ∈ V)
3 lcoop.b . . . . . 6 𝐵 = (Base‘𝑀)
43pweqi 4139 . . . . 5 𝒫 𝐵 = 𝒫 (Base‘𝑀)
54eleq2i 2696 . . . 4 (𝑉 ∈ 𝒫 𝐵𝑉 ∈ 𝒫 (Base‘𝑀))
65biimpi 206 . . 3 (𝑉 ∈ 𝒫 𝐵𝑉 ∈ 𝒫 (Base‘𝑀))
76adantl 482 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → 𝑉 ∈ 𝒫 (Base‘𝑀))
8 fvex 6160 . . . 4 (Base‘𝑀) ∈ V
93, 8eqeltri 2700 . . 3 𝐵 ∈ V
10 rabexg 4777 . . 3 (𝐵 ∈ V → {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ∈ V)
119, 10mp1i 13 . 2 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ∈ V)
12 fveq2 6150 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
1312, 3syl6eqr 2678 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
1413adantr 481 . . . 4 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘𝑚) = 𝐵)
15 fveq2 6150 . . . . . . . . 9 (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
1615fveq2d 6154 . . . . . . . 8 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
1716adantr 481 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘𝑀)))
18 lcoop.r . . . . . . . 8 𝑅 = (Base‘𝑆)
19 lcoop.s . . . . . . . . 9 𝑆 = (Scalar‘𝑀)
2019fveq2i 6153 . . . . . . . 8 (Base‘𝑆) = (Base‘(Scalar‘𝑀))
2118, 20eqtri 2648 . . . . . . 7 𝑅 = (Base‘(Scalar‘𝑀))
2217, 21syl6eqr 2678 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (Base‘(Scalar‘𝑚)) = 𝑅)
23 simpr 477 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → 𝑣 = 𝑉)
2422, 23oveq12d 6623 . . . . 5 ((𝑚 = 𝑀𝑣 = 𝑉) → ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣) = (𝑅𝑚 𝑉))
2515fveq2d 6154 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘(Scalar‘𝑀)))
2619a1i 11 . . . . . . . . . . 11 (𝑚 = 𝑀𝑆 = (Scalar‘𝑀))
2726eqcomd 2632 . . . . . . . . . 10 (𝑚 = 𝑀 → (Scalar‘𝑀) = 𝑆)
2827fveq2d 6154 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑀)) = (0g𝑆))
2925, 28eqtrd 2660 . . . . . . . 8 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g𝑆))
3029adantr 481 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (0g‘(Scalar‘𝑚)) = (0g𝑆))
3130breq2d 4630 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑠 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑠 finSupp (0g𝑆)))
32 fveq2 6150 . . . . . . . . 9 (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
3332adantr 481 . . . . . . . 8 ((𝑚 = 𝑀𝑣 = 𝑉) → ( linC ‘𝑚) = ( linC ‘𝑀))
34 eqidd 2627 . . . . . . . 8 ((𝑚 = 𝑀𝑣 = 𝑉) → 𝑠 = 𝑠)
3533, 34, 23oveq123d 6626 . . . . . . 7 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑠( linC ‘𝑚)𝑣) = (𝑠( linC ‘𝑀)𝑉))
3635eqeq2d 2636 . . . . . 6 ((𝑚 = 𝑀𝑣 = 𝑉) → (𝑐 = (𝑠( linC ‘𝑚)𝑣) ↔ 𝑐 = (𝑠( linC ‘𝑀)𝑉)))
3731, 36anbi12d 746 . . . . 5 ((𝑚 = 𝑀𝑣 = 𝑉) → ((𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣)) ↔ (𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))))
3824, 37rexeqbidv 3147 . . . 4 ((𝑚 = 𝑀𝑣 = 𝑉) → (∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣)) ↔ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))))
3914, 38rabeqbidv 3186 . . 3 ((𝑚 = 𝑀𝑣 = 𝑉) → {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))} = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
4012pweqd 4140 . . 3 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
41 df-lco 41458 . . 3 LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))})
4239, 40, 41ovmpt2x 6743 . 2 ((𝑀 ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))} ∈ V) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
432, 7, 11, 42syl3anc 1323 1 ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wrex 2913  {crab 2916  Vcvv 3191  𝒫 cpw 4135   class class class wbr 4618  cfv 5850  (class class class)co 6605  𝑚 cmap 7803   finSupp cfsupp 8220  Basecbs 15776  Scalarcsca 15860  0gc0g 16016   linC clinc 41455   LinCo clinco 41456
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-lco 41458
This theorem is referenced by:  lcoval  41463  lco0  41478
  Copyright terms: Public domain W3C validator