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Theorem lshpset 34846
 Description: The set of all hyperplanes of a left module or left vector space. The vector is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)
Hypotheses
Ref Expression
lshpset.v
lshpset.n
lshpset.s
lshpset.h LSHyp
Assertion
Ref Expression
lshpset
Distinct variable groups:   ,   ,   ,,
Allowed substitution hints:   ()   (,)   (,)   ()   (,)

Proof of Theorem lshpset
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 lshpset.h . 2 LSHyp
2 elex 3118 . . 3
3 fveq2 5872 . . . . . 6
4 lshpset.s . . . . . 6
53, 4syl6eqr 2516 . . . . 5
6 fveq2 5872 . . . . . . . 8
7 lshpset.v . . . . . . . 8
86, 7syl6eqr 2516 . . . . . . 7
98neeq2d 2735 . . . . . 6
10 fveq2 5872 . . . . . . . . . 10
11 lshpset.n . . . . . . . . . 10
1210, 11syl6eqr 2516 . . . . . . . . 9
1312fveq1d 5874 . . . . . . . 8
1413, 8eqeq12d 2479 . . . . . . 7
158, 14rexeqbidv 3069 . . . . . 6
169, 15anbi12d 710 . . . . 5
175, 16rabeqbidv 3104 . . . 4
18 df-lshyp 34845 . . . 4 LSHyp
19 fvex 5882 . . . . . 6
204, 19eqeltri 2541 . . . . 5
2120rabex 4607 . . . 4
2217, 18, 21fvmpt 5956 . . 3 LSHyp
232, 22syl 16 . 2 LSHyp
241, 23syl5eq 2510 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819   wne 2652  wrex 2808  crab 2811  cvv 3109   cun 3469  csn 4032  cfv 5594  cbs 14644  clss 17705  clspn 17744  LSHypclsh 34843 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-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  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-lshyp 34845 This theorem is referenced by:  islshp  34847
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