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Theorem stoweidlem6 42176
Description: Lemma for stoweid 42233: 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 1132 . 2 ((𝜑𝐹𝐴𝐺𝐴) → 𝐺𝐴)
2 eleq1 2905 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐴𝐺𝐴))
323anbi3d 1435 . . . 4 (𝑔 = 𝐺 → ((𝜑𝐹𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝐺𝐴)))
4 stoweidlem6.2 . . . . . 6 𝑡 𝑔 = 𝐺
5 fveq1 6668 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑡) = (𝐺𝑡))
65oveq2d 7166 . . . . . . 7 (𝑔 = 𝐺 → ((𝐹𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝐺𝑡)))
76adantr 481 . . . . . 6 ((𝑔 = 𝐺𝑡𝑇) → ((𝐹𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝐺𝑡)))
84, 7mpteq2da 5157 . . . . 5 (𝑔 = 𝐺 → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))))
98eleq1d 2902 . . . 4 (𝑔 = 𝐺 → ((𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴))
103, 9imbi12d 346 . . 3 (𝑔 = 𝐺 → (((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴)))
11 simp2 1131 . . . 4 ((𝜑𝐹𝐴𝑔𝐴) → 𝐹𝐴)
12 eleq1 2905 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
13123anbi2d 1434 . . . . . 6 (𝑓 = 𝐹 → ((𝜑𝑓𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝑔𝐴)))
14 stoweidlem6.1 . . . . . . . 8 𝑡 𝑓 = 𝐹
15 fveq1 6668 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑡) = (𝐹𝑡))
1615oveq1d 7165 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝑔𝑡)))
1716adantr 481 . . . . . . . 8 ((𝑓 = 𝐹𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝑔𝑡)))
1814, 17mpteq2da 5157 . . . . . . 7 (𝑓 = 𝐹 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))))
1918eleq1d 2902 . . . . . 6 (𝑓 = 𝐹 → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴))
2013, 19imbi12d 346 . . . . 5 (𝑓 = 𝐹 → (((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴)))
21 stoweidlem6.3 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
2220, 21vtoclg 3573 . . . 4 (𝐹𝐴 → ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴))
2311, 22mpcom 38 . . 3 ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴)
2410, 23vtoclg 3573 . 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 1081   = wceq 1530  wnf 1777  wcel 2107  cmpt 5143  cfv 6354  (class class class)co 7150   · cmul 10536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-iota 6313  df-fv 6362  df-ov 7153
This theorem is referenced by:  stoweidlem19  42189  stoweidlem22  42192  stoweidlem32  42202  stoweidlem36  42206
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