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Theorem stoweidlem6 39556
Description: Lemma for stoweid 39613: 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 1061 . 2 ((𝜑𝐹𝐴𝐺𝐴) → 𝐺𝐴)
2 eleq1 2686 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐴𝐺𝐴))
323anbi3d 1402 . . . 4 (𝑔 = 𝐺 → ((𝜑𝐹𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝐺𝐴)))
4 stoweidlem6.2 . . . . . 6 𝑡 𝑔 = 𝐺
5 fveq1 6152 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑡) = (𝐺𝑡))
65oveq2d 6626 . . . . . . 7 (𝑔 = 𝐺 → ((𝐹𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝐺𝑡)))
76adantr 481 . . . . . 6 ((𝑔 = 𝐺𝑡𝑇) → ((𝐹𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝐺𝑡)))
84, 7mpteq2da 4708 . . . . 5 (𝑔 = 𝐺 → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))))
98eleq1d 2683 . . . 4 (𝑔 = 𝐺 → ((𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴))
103, 9imbi12d 334 . . 3 (𝑔 = 𝐺 → (((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴)))
11 simp2 1060 . . . 4 ((𝜑𝐹𝐴𝑔𝐴) → 𝐹𝐴)
12 eleq1 2686 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
13123anbi2d 1401 . . . . . 6 (𝑓 = 𝐹 → ((𝜑𝑓𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝑔𝐴)))
14 stoweidlem6.1 . . . . . . . 8 𝑡 𝑓 = 𝐹
15 fveq1 6152 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑡) = (𝐹𝑡))
1615oveq1d 6625 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝑔𝑡)))
1716adantr 481 . . . . . . . 8 ((𝑓 = 𝐹𝑡𝑇) → ((𝑓𝑡) · (𝑔𝑡)) = ((𝐹𝑡) · (𝑔𝑡)))
1814, 17mpteq2da 4708 . . . . . . 7 (𝑓 = 𝐹 → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))))
1918eleq1d 2683 . . . . . 6 (𝑓 = 𝐹 → ((𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴))
2013, 19imbi12d 334 . . . . 5 (𝑓 = 𝐹 → (((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴)))
21 stoweidlem6.3 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
2220, 21vtoclg 3255 . . . 4 (𝐹𝐴 → ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴))
2311, 22mpcom 38 . . 3 ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝑔𝑡))) ∈ 𝐴)
2410, 23vtoclg 3255 . 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 1036   = wceq 1480  wnf 1705  wcel 1987  cmpt 4678  cfv 5852  (class class class)co 6610   · cmul 9893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-iota 5815  df-fv 5860  df-ov 6613
This theorem is referenced by:  stoweidlem19  39569  stoweidlem22  39572  stoweidlem32  39582  stoweidlem36  39586
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