Theorem scafvalg 14256

Description: The scalar multiplication operation as 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)
scaffval.s	⊢ · = ( ·𝑠 ‘𝑊)

Assertion

Ref	Expression
scafvalg	⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = (𝑋 · 𝑌))

Proof of Theorem scafvalg

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

Step	Hyp	Ref	Expression
1		scaffval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
2		scaffval.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
3		scaffval.k	. . . 4 ⊢ 𝐾 = (Base‘𝐹)
4		scaffval.a	. . . 4 ⊢ ∙ = ( ·sf ‘𝑊)
5		scaffval.s	. . . 4 ⊢ · = ( ·𝑠 ‘𝑊)
6	1, 2, 3, 4, 5	scaffvalg 14255	. . 3 ⊢ (𝑊 ∈ 𝑉 → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
7	6	3ad2ant1 1042	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦)))
8		oveq12 6003	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 · 𝑦) = (𝑋 · 𝑌))
9	8	adantl 277	. 2 ⊢ (((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥 · 𝑦) = (𝑋 · 𝑌))
10		simp2 1022	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐾)
11		simp3 1023	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
12		vscaslid 13182	. . . . . 6 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
13	12	slotex 13045	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → ( ·𝑠 ‘𝑊) ∈ V)
14	5, 13	eqeltrid 2316	. . . 4 ⊢ (𝑊 ∈ 𝑉 → · ∈ V)
15	14	3ad2ant1 1042	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → · ∈ V)
16		ovexg 6028	. . 3 ⊢ ((𝑋 ∈ 𝐾 ∧ · ∈ V ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ V)
17	10, 15, 11, 16	syl3anc 1271	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ V)
18	7, 9, 10, 11, 17	ovmpod 6123	1 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = (𝑋 · 𝑌))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ‘cfv 5314  (class class class)co 5994   ∈ cmpo 5996  Basecbs 13018  Scalarcsca 13099   ·_𝑠 cvsca 13100   ·_sf cscaf 14237

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-ndx 13021  df-slot 13022  df-base 13024  df-sca 13112  df-vsca 13113  df-scaf 14239

This theorem is referenced by:  lmodfopne  14275