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Theorem stoweidlem6 43014
 Description: Lemma for stoweid 43071: 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 1135 . 2 ((𝜑𝐹𝐴𝐺𝐴) → 𝐺𝐴)
2 eleq1 2839 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐴𝐺𝐴))
323anbi3d 1439 . . . 4 (𝑔 = 𝐺 → ((𝜑𝐹𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝐺𝐴)))
4 stoweidlem6.2 . . . . . 6 𝑡 𝑔 = 𝐺
5 fveq1 6657 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑡) = (𝐺𝑡))
65oveq2d 7166 . . . . . . 7 (𝑔 = 𝐺 → ((𝐹𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝐺𝑡)))
76adantr 484 . . . . . 6 ((𝑔 = 𝐺𝑡𝑇) → ((𝐹𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝐺𝑡)))
84, 7mpteq2da 5126 . . . . 5 (𝑔 = 𝐺 → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))))
98eleq1d 2836 . . . 4 (𝑔 = 𝐺 → ((𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴))
103, 9imbi12d 348 . . 3 (𝑔 = 𝐺 → (((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴)))
11 simp2 1134 . . . 4 ((𝜑𝐹𝐴𝑔𝐴) → 𝐹𝐴)
12 eleq1 2839 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
13123anbi2d 1438 . . . . . 6 (𝑓 = 𝐹 → ((𝜑𝑓𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝑔𝐴)))
14 stoweidlem6.1 . . . . . . . 8 𝑡 𝑓 = 𝐹
15 fveq1 6657 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑡) = (𝐹𝑡))
1615oveq1d 7165 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝑔𝑡)))
1716adantr 484 . . . . . . . 8 ((𝑓 = 𝐹𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝑔𝑡)))
1814, 17mpteq2da 5126 . . . . . . 7 (𝑓 = 𝐹 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))))
1918eleq1d 2836 . . . . . 6 (𝑓 = 𝐹 → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴))
2013, 19imbi12d 348 . . . . 5 (𝑓 = 𝐹 → (((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴)))
21 stoweidlem6.3 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
2220, 21vtoclg 3485 . . . 4 (𝐹𝐴 → ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴))
2311, 22mpcom 38 . . 3 ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴)
2410, 23vtoclg 3485 . 2 (𝐺𝐴 → ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴))
251, 24mpcom 38 1 ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111   ↦ cmpt 5112  ‘cfv 6335  (class class class)co 7150   · cmul 10580 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-iota 6294  df-fv 6343  df-ov 7153 This theorem is referenced by:  stoweidlem19  43027  stoweidlem22  43030  stoweidlem32  43040  stoweidlem36  43044
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