Theorem lmodscaf 14584

Description: The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
scaffval.b	⊢ 𝐵 = (Base‘𝑊)
scaffval.f	⊢ 𝐹 = (Scalar‘𝑊)
scaffval.k	⊢ 𝐾 = (Base‘𝐹)
scaffval.a	⊢ ∙ = ( ·sf ‘𝑊)

Assertion

Ref	Expression
lmodscaf	⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵)

Proof of Theorem lmodscaf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
2		scaffval.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
3		eqid 2234	. . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
4		scaffval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
5	1, 2, 3, 4	lmodvscl 14579	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
6	5	3expb 1231	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
7	6	ralrimivva 2626	. . 3 ⊢ (𝑊 ∈ LMod → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵)
8		eqid 2234	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦))
9	8	fmpo 6410	. . 3 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐵 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝐵 ↔ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)):(𝐾 × 𝐵)⟶𝐵)
10	7, 9	sylib 122	. 2 ⊢ (𝑊 ∈ LMod → (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)):(𝐾 × 𝐵)⟶𝐵)
11		scaffval.a	. . . 4 ⊢ ∙ = ( ·sf ‘𝑊)
12	1, 2, 4, 11, 3	scaffvalg 14580	. . 3 ⊢ (𝑊 ∈ LMod → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)))
13	12	feq1d 5500	. 2 ⊢ (𝑊 ∈ LMod → ( ∙ :(𝐾 × 𝐵)⟶𝐵 ↔ (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥( ·𝑠 ‘𝑊)𝑦)):(𝐾 × 𝐵)⟶𝐵))
14	10, 13	mpbird 167	1 ⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  ∀wral 2522   × cxp 4752  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058   ∈ cmpo 6060  Basecbs 13296  Scalarcsca 13377   ·_𝑠 cvsca 13378  LModclmod 14561   ·_sf cscaf 14562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-mulr 13388  df-sca 13390  df-vsca 13391  df-lmod 14563  df-scaf 14564

This theorem is referenced by: (None)