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Theorem iscph 19138
 Description: A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complexes, with a norm defined (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
iscph.v
iscph.h
iscph.n
iscph.f Scalar
iscph.k
Assertion
Ref Expression
iscph NrmMod flds
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()   ()

Proof of Theorem iscph
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3532 . . . . 5 NrmMod NrmMod
21anbi1i 678 . . . 4 NrmMod flds NrmMod flds
3 df-3an 939 . . . 4 NrmMod flds NrmMod flds
42, 3bitr4i 245 . . 3 NrmMod flds NrmMod flds
54anbi1i 678 . 2 NrmMod flds NrmMod flds
6 fvex 5745 . . . . . 6 Scalar
76a1i 11 . . . . 5 Scalar
8 fvex 5745 . . . . . . 7
98a1i 11 . . . . . 6 Scalar
10 simplr 733 . . . . . . . . . 10 Scalar Scalar
11 simpll 732 . . . . . . . . . . . 12 Scalar
1211fveq2d 5735 . . . . . . . . . . 11 Scalar Scalar Scalar
13 iscph.f . . . . . . . . . . 11 Scalar
1412, 13syl6eqr 2488 . . . . . . . . . 10 Scalar Scalar
1510, 14eqtrd 2470 . . . . . . . . 9 Scalar
16 simpr 449 . . . . . . . . . . 11 Scalar
1715fveq2d 5735 . . . . . . . . . . . 12 Scalar
18 iscph.k . . . . . . . . . . . 12
1917, 18syl6eqr 2488 . . . . . . . . . . 11 Scalar
2016, 19eqtrd 2470 . . . . . . . . . 10 Scalar
2120oveq2d 6100 . . . . . . . . 9 Scalar flds flds
2215, 21eqeq12d 2452 . . . . . . . 8 Scalar flds flds
2320ineq1d 3543 . . . . . . . . . 10 Scalar
2423imaeq2d 5206 . . . . . . . . 9 Scalar
2524, 20sseq12d 3379 . . . . . . . 8 Scalar
2611fveq2d 5735 . . . . . . . . . 10 Scalar
27 iscph.n . . . . . . . . . 10
2826, 27syl6eqr 2488 . . . . . . . . 9 Scalar
2911fveq2d 5735 . . . . . . . . . . 11 Scalar
30 iscph.v . . . . . . . . . . 11
3129, 30syl6eqr 2488 . . . . . . . . . 10 Scalar
3211fveq2d 5735 . . . . . . . . . . . . 13 Scalar
33 iscph.h . . . . . . . . . . . . 13
3432, 33syl6eqr 2488 . . . . . . . . . . . 12 Scalar
3534oveqd 6101 . . . . . . . . . . 11 Scalar
3635fveq2d 5735 . . . . . . . . . 10 Scalar
3731, 36mpteq12dv 4290 . . . . . . . . 9 Scalar
3828, 37eqeq12d 2452 . . . . . . . 8 Scalar
3922, 25, 383anbi123d 1255 . . . . . . 7 Scalar flds flds
40 3anass 941 . . . . . . 7 flds flds
4139, 40syl6bb 254 . . . . . 6 Scalar flds flds
429, 41sbcied 3199 . . . . 5 Scalar flds flds
437, 42sbcied 3199 . . . 4 Scalar flds flds
44 df-cph 19136 . . . 4 NrmMod Scalar flds
4543, 44elrab2 3096 . . 3 NrmMod flds
46 anass 632 . . 3 NrmMod flds NrmMod flds
4745, 46bitr4i 245 . 2 NrmMod flds
48 3anass 941 . 2 NrmMod flds NrmMod flds
495, 47, 483bitr4i 270 1 NrmMod flds
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  cvv 2958  wsbc 3163   cin 3321   wss 3322   cmpt 4269  cima 4884  cfv 5457  (class class class)co 6084  cc0 8995   cpnf 9122  cico 10923  csqr 12043  cbs 13474   ↾s cress 13475  Scalarcsca 13537  cip 13539  ℂfldccnfld 16708  cphl 16860  cnm 18629  NrmModcnlm 18633  ccph 19134 This theorem is referenced by:  cphphl  19139  cphnlm  19140  cphsca  19147  cphsqrcl  19152  cphnmfval  19160  tchcph  19199 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4341 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-ov 6087  df-cph 19136
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