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Theorem islininds 32120
 Description: The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
islininds.b
islininds.z
islininds.r Scalar
islininds.e
islininds.0
Assertion
Ref Expression
islininds linIndS finSupp linC
Distinct variable groups:   ,   ,,   ,,
Allowed substitution hints:   (,)   (,)   ()   (,)   (,)   (,)   (,)

Proof of Theorem islininds
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . 4
2 fveq2 5864 . . . . . . 7
3 islininds.b . . . . . . 7
42, 3syl6eqr 2526 . . . . . 6
54adantl 466 . . . . 5
65pweqd 4015 . . . 4
71, 6eleq12d 2549 . . 3
8 fveq2 5864 . . . . . . . . 9 Scalar Scalar
9 islininds.r . . . . . . . . 9 Scalar
108, 9syl6eqr 2526 . . . . . . . 8 Scalar
1110fveq2d 5868 . . . . . . 7 Scalar
12 islininds.e . . . . . . 7
1311, 12syl6eqr 2526 . . . . . 6 Scalar
1413adantl 466 . . . . 5 Scalar
1514, 1oveq12d 6300 . . . 4 Scalar
168adantl 466 . . . . . . . . . 10 Scalar Scalar
1716, 9syl6eqr 2526 . . . . . . . . 9 Scalar
1817fveq2d 5868 . . . . . . . 8 Scalar
19 islininds.0 . . . . . . . 8
2018, 19syl6eqr 2526 . . . . . . 7 Scalar
2120breq2d 4459 . . . . . 6 finSupp Scalar finSupp
22 fveq2 5864 . . . . . . . . 9 linC linC
2322adantl 466 . . . . . . . 8 linC linC
24 eqidd 2468 . . . . . . . 8
2523, 24, 1oveq123d 6303 . . . . . . 7 linC linC
26 fveq2 5864 . . . . . . . . 9
2726adantl 466 . . . . . . . 8
28 islininds.z . . . . . . . 8
2927, 28syl6eqr 2526 . . . . . . 7
3025, 29eqeq12d 2489 . . . . . 6 linC linC
3121, 30anbi12d 710 . . . . 5 finSupp Scalar linC finSupp linC
3210fveq2d 5868 . . . . . . . . 9 Scalar
3332, 19syl6eqr 2526 . . . . . . . 8 Scalar
3433adantl 466 . . . . . . 7 Scalar
3534eqeq2d 2481 . . . . . 6 Scalar
361, 35raleqbidv 3072 . . . . 5 Scalar
3731, 36imbi12d 320 . . . 4 finSupp Scalar linC Scalar finSupp linC
3815, 37raleqbidv 3072 . . 3 Scalar finSupp Scalar linC Scalar finSupp linC
397, 38anbi12d 710 . 2 Scalar finSupp Scalar linC Scalar finSupp linC
40 df-lininds 32116 . 2 linIndS Scalar finSupp Scalar linC Scalar
4139, 40brabga 4761 1 linIndS finSupp linC
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  wral 2814  cpw 4010   class class class wbr 4447  cfv 5586  (class class class)co 6282   cmap 7417   finSupp cfsupp 7825  cbs 14483  Scalarcsca 14551  c0g 14688   linC clinc 32078   linIndS clininds 32114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-iota 5549  df-fv 5594  df-ov 6285  df-lininds 32116 This theorem is referenced by:  linindsi  32121  islinindfis  32123  islindeps  32127  lindslininds  32138  linds0  32139  lindsrng01  32142  snlindsntor  32145
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