Theorem isclmi 25217

Description: Reverse direction of isclm 25204. (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 1154	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ LMod)
2		simp2 1155	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂfld ↾s 𝐾))
3		eqid 2763	. . . . . . 7 ⊢ (ℂfld ↾s 𝐾) = (ℂfld ↾s 𝐾)
4	3	subrgbas 20667	. . . . . 6 ⊢ (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 = (Base‘(ℂfld ↾s 𝐾)))
5	4	3ad2ant3 1153	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘(ℂfld ↾s 𝐾)))
6	2	fveq2d 6887	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) = (Base‘(ℂfld ↾s 𝐾)))
7	5, 6	eqtr4d 2801	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘𝐹))
8	7	oveq2d 7428	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (ℂfld ↾s 𝐾) = (ℂfld ↾s (Base‘𝐹)))
9	2, 8	eqtrd 2798	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂfld ↾s (Base‘𝐹)))
10		simp3 1156	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 ∈ (SubRing‘ℂfld))
11	7, 10	eqeltrrd 2864	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) ∈ (SubRing‘ℂfld))
12		clm0.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
13		eqid 2763	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
14	12, 13	isclm 25204	. 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld)))
15	1, 9, 11, 14	syl3anbrc 1362	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ‘cfv 6538  (class class class)co 7412  Basecbs 17270   ↾_s cress 17291  Scalarcsca 17314  SubRingcsubrg 20655  LModclmod 20962  ℂ_fldccnfld 21503  ℂModcclm 25202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-1cn 11159  ax-addcl 11161

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-nn 12235  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-subg 19190  df-ring 20318  df-subrg 20656  df-clm 25203

This theorem is referenced by:  zlmclm  25252  cnstrcvs  25281  cncvs  25285  recvs  25286  qcvs  25287  zclmncvs  25288