Theorem isclmi 25033

Description: Reverse direction of isclm 25020. (Contributed by Mario Carneiro, 30-Oct-2015.)

Hypothesis

Ref	Expression
clm0.f	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
isclmi	⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)

Proof of Theorem isclmi

Step	Hyp	Ref	Expression
1		simp1 1136	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ LMod)
2		simp2 1137	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂfld ↾s 𝐾))
3		eqid 2736	. . . . . . 7 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾)
4	3	subrgbas 20514	. . . . . 6 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 = (Base‘(ℂfld ↾s 𝐾)))
5	4	3ad2ant3 1135	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘(ℂfld ↾s 𝐾)))
6	2	fveq2d 6838	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) = (Base‘(ℂfld ↾s 𝐾)))
7	5, 6	eqtr4d 2774	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘𝐹))
8	7	oveq2d 7374	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (ℂfld ↾s 𝐾) = (ℂfld ↾s (Base‘𝐹)))
9	2, 8	eqtrd 2771	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂfld ↾s (Base‘𝐹)))
10		simp3 1138	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 ∈ (SubRing‘ℂfld))
11	7, 10	eqeltrrd 2837	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) ∈ (SubRing‘ℂfld))
12		clm0.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
13		eqid 2736	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
14	12, 13	isclm 25020	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld)))
15	1, 9, 11, 14	syl3anbrc 1344	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ‘cfv 6492  (class class class)co 7358  Basecbs 17136   ↾_s cress 17157  Scalarcsca 17180  SubRingcsubrg 20502  LModclmod 20811  ℂ_fldccnfld 21309  ℂModcclm 25018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-1cn 11084  ax-addcl 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-nn 12146  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-subg 19053  df-ring 20170  df-subrg 20503  df-clm 25019

This theorem is referenced by:  zlmclm  25068  cnstrcvs  25097  cncvs  25101  recvs  25102  qcvs  25103  zclmncvs  25104