Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem6 Structured version   Visualization version   GIF version

Theorem stoweidlem6 38899
 Description: Lemma for stoweid 38956: two class variables replace two setvar variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem6.1 𝑡 𝑓 = 𝐹
stoweidlem6.2 𝑡 𝑔 = 𝐺
stoweidlem6.3 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
Assertion
Ref Expression
stoweidlem6 ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴)
Distinct variable groups:   𝑓,𝑔,𝑡   𝐴,𝑓,𝑔   𝑓,𝐹,𝑔   𝑇,𝑓,𝑔   𝜑,𝑓,𝑔   𝑔,𝐺
Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑡)   𝑇(𝑡)   𝐹(𝑡)   𝐺(𝑡,𝑓)

Proof of Theorem stoweidlem6
StepHypRef Expression
1 simp3 1056 . 2 ((𝜑𝐹𝐴𝐺𝐴) → 𝐺𝐴)
2 eleq1 2676 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐴𝐺𝐴))
323anbi3d 1397 . . . 4 (𝑔 = 𝐺 → ((𝜑𝐹𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝐺𝐴)))
4 stoweidlem6.2 . . . . . 6 𝑡 𝑔 = 𝐺
5 fveq1 6102 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑡) = (𝐺𝑡))
65oveq2d 6565 . . . . . . 7 (𝑔 = 𝐺 → ((𝐹𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝐺𝑡)))
76adantr 480 . . . . . 6 ((𝑔 = 𝐺𝑡𝑇) → ((𝐹𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝐺𝑡)))
84, 7mpteq2da 4671 . . . . 5 (𝑔 = 𝐺 → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))))
98eleq1d 2672 . . . 4 (𝑔 = 𝐺 → ((𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴))
103, 9imbi12d 333 . . 3 (𝑔 = 𝐺 → (((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴)))
11 simp2 1055 . . . 4 ((𝜑𝐹𝐴𝑔𝐴) → 𝐹𝐴)
12 eleq1 2676 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
13123anbi2d 1396 . . . . . 6 (𝑓 = 𝐹 → ((𝜑𝑓𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝑔𝐴)))
14 stoweidlem6.1 . . . . . . . 8 𝑡 𝑓 = 𝐹
15 fveq1 6102 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑡) = (𝐹𝑡))
1615oveq1d 6564 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝑔𝑡)))
1716adantr 480 . . . . . . . 8 ((𝑓 = 𝐹𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝑔𝑡)))
1814, 17mpteq2da 4671 . . . . . . 7 (𝑓 = 𝐹 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))))
1918eleq1d 2672 . . . . . 6 (𝑓 = 𝐹 → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴))
2013, 19imbi12d 333 . . . . 5 (𝑓 = 𝐹 → (((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴)))
21 stoweidlem6.3 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
2220, 21vtoclg 3239 . . . 4 (𝐹𝐴 → ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴))
2311, 22mpcom 37 . . 3 ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴)
2410, 23vtoclg 3239 . 2 (𝐺𝐴 → ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴))
251, 24mpcom 37 1 ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   · cmul 9820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  stoweidlem19  38912  stoweidlem22  38915  stoweidlem32  38925  stoweidlem36  38929
 Copyright terms: Public domain W3C validator