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Theorem stoweidlem8 42278
Description: Lemma for stoweid 42333: two class variables replace two setvar variables, for the sum of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem8.1 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem8.2 𝑡𝐹
stoweidlem8.3 𝑡𝐺
Assertion
Ref Expression
stoweidlem8 ((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 stoweidlem8
StepHypRef Expression
1 simp3 1132 . 2 ((𝜑𝐹𝐴𝐺𝐴) → 𝐺𝐴)
2 eleq1 2898 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐴𝐺𝐴))
323anbi3d 1435 . . . 4 (𝑔 = 𝐺 → ((𝜑𝐹𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝐺𝐴)))
4 stoweidlem8.3 . . . . . . 7 𝑡𝐺
54nfeq2 2993 . . . . . 6 𝑡 𝑔 = 𝐺
6 fveq1 6662 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑡) = (𝐺𝑡))
76oveq2d 7164 . . . . . . 7 (𝑔 = 𝐺 → ((𝐹𝑡) + (𝑔𝑡)) = ((𝐹𝑡) + (𝐺𝑡)))
87adantr 483 . . . . . 6 ((𝑔 = 𝐺𝑡𝑇) → ((𝐹𝑡) + (𝑔𝑡)) = ((𝐹𝑡) + (𝐺𝑡)))
95, 8mpteq2da 5151 . . . . 5 (𝑔 = 𝐺 → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))))
109eleq1d 2895 . . . 4 (𝑔 = 𝐺 → ((𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))) ∈ 𝐴))
113, 10imbi12d 347 . . 3 (𝑔 = 𝐺 → (((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))) ∈ 𝐴)))
12 simp2 1131 . . . 4 ((𝜑𝐹𝐴𝑔𝐴) → 𝐹𝐴)
13 eleq1 2898 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
14133anbi2d 1434 . . . . . 6 (𝑓 = 𝐹 → ((𝜑𝑓𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝑔𝐴)))
15 stoweidlem8.2 . . . . . . . . 9 𝑡𝐹
1615nfeq2 2993 . . . . . . . 8 𝑡 𝑓 = 𝐹
17 fveq1 6662 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑡) = (𝐹𝑡))
1817oveq1d 7163 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑡) + (𝑔𝑡)) = ((𝐹𝑡) + (𝑔𝑡)))
1918adantr 483 . . . . . . . 8 ((𝑓 = 𝐹𝑡𝑇) → ((𝑓𝑡) + (𝑔𝑡)) = ((𝐹𝑡) + (𝑔𝑡)))
2016, 19mpteq2da 5151 . . . . . . 7 (𝑓 = 𝐹 → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))))
2120eleq1d 2895 . . . . . 6 (𝑓 = 𝐹 → ((𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴))
2214, 21imbi12d 347 . . . . 5 (𝑓 = 𝐹 → (((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴)))
23 stoweidlem8.1 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
2422, 23vtoclg 3566 . . . 4 (𝐹𝐴 → ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴))
2512, 24mpcom 38 . . 3 ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴)
2611, 25vtoclg 3566 . 2 (𝐺𝐴 → ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))) ∈ 𝐴))
271, 26mpcom 38 1 ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  wnfc 2959  cmpt 5137  cfv 6348  (class class class)co 7148   + caddc 10532
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 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-iota 6307  df-fv 6356  df-ov 7151
This theorem is referenced by:  stoweidlem20  42290  stoweidlem21  42291  stoweidlem22  42292  stoweidlem23  42293
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