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Theorem lcoop 33114
 Description: A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lcoop.b
lcoop.s Scalar
lcoop.r
Assertion
Ref Expression
lcoop LinCo finSupp linC
Distinct variable groups:   ,   ,,   ,,   ,,
Allowed substitution hints:   ()   (,)   (,)

Proof of Theorem lcoop
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3118 . . 3
3 lcoop.b . . . . . 6
43pweqi 4019 . . . . 5
54eleq2i 2535 . . . 4
65biimpi 194 . . 3
8 fvex 5882 . . . 4
93, 8eqeltri 2541 . . 3
10 rabexg 4606 . . 3 finSupp linC
119, 10mp1i 12 . 2 finSupp linC
12 fveq2 5872 . . . . . 6
1312, 3syl6eqr 2516 . . . . 5
1413adantr 465 . . . 4
15 fveq2 5872 . . . . . . . . 9 Scalar Scalar
1615fveq2d 5876 . . . . . . . 8 Scalar Scalar
1716adantr 465 . . . . . . 7 Scalar Scalar
18 lcoop.r . . . . . . . 8
19 lcoop.s . . . . . . . . 9 Scalar
2019fveq2i 5875 . . . . . . . 8 Scalar
2118, 20eqtri 2486 . . . . . . 7 Scalar
2217, 21syl6eqr 2516 . . . . . 6 Scalar
23 simpr 461 . . . . . 6
2422, 23oveq12d 6314 . . . . 5 Scalar
2515fveq2d 5876 . . . . . . . . 9 Scalar Scalar
2619a1i 11 . . . . . . . . . . 11 Scalar
2726eqcomd 2465 . . . . . . . . . 10 Scalar
2827fveq2d 5876 . . . . . . . . 9 Scalar
2925, 28eqtrd 2498 . . . . . . . 8 Scalar
3029adantr 465 . . . . . . 7 Scalar
3130breq2d 4468 . . . . . 6 finSupp Scalar finSupp
32 fveq2 5872 . . . . . . . . 9 linC linC
3332adantr 465 . . . . . . . 8 linC linC
34 eqidd 2458 . . . . . . . 8
3533, 34, 23oveq123d 6317 . . . . . . 7 linC linC
3635eqeq2d 2471 . . . . . 6 linC linC
3731, 36anbi12d 710 . . . . 5 finSupp Scalar linC finSupp linC
3824, 37rexeqbidv 3069 . . . 4 Scalar finSupp Scalar linC finSupp linC
3914, 38rabeqbidv 3104 . . 3 Scalar finSupp Scalar linC finSupp linC
4012pweqd 4020 . . 3
41 df-lco 33110 . . 3 LinCo Scalar finSupp Scalar linC
4239, 40, 41ovmpt2x 6430 . 2 finSupp linC LinCo finSupp linC
432, 7, 11, 42syl3anc 1228 1 LinCo finSupp linC
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819  wrex 2808  crab 2811  cvv 3109  cpw 4015   class class class wbr 4456  cfv 5594  (class class class)co 6296   cmap 7438   finSupp cfsupp 7847  cbs 14643  Scalarcsca 14714  c0g 14856   linC clinc 33107   LinCo clinco 33108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-lco 33110 This theorem is referenced by:  lcoval  33115  lco0  33130
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