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Theorem stoweidlem8 40968
Description: Lemma for stoweid 41023: 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 1169 . 2 ((𝜑𝐹𝐴𝐺𝐴) → 𝐺𝐴)
2 eleq1 2866 . . . . 5 (𝑔 = 𝐺 → (𝑔𝐴𝐺𝐴))
323anbi3d 1567 . . . 4 (𝑔 = 𝐺 → ((𝜑𝐹𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝐺𝐴)))
4 stoweidlem8.3 . . . . . . 7 𝑡𝐺
54nfeq2 2957 . . . . . 6 𝑡 𝑔 = 𝐺
6 fveq1 6410 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔𝑡) = (𝐺𝑡))
76oveq2d 6894 . . . . . . 7 (𝑔 = 𝐺 → ((𝐹𝑡) + (𝑔𝑡)) = ((𝐹𝑡) + (𝐺𝑡)))
87adantr 473 . . . . . 6 ((𝑔 = 𝐺𝑡𝑇) → ((𝐹𝑡) + (𝑔𝑡)) = ((𝐹𝑡) + (𝐺𝑡)))
95, 8mpteq2da 4936 . . . . 5 (𝑔 = 𝐺 → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))))
109eleq1d 2863 . . . 4 (𝑔 = 𝐺 → ((𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))) ∈ 𝐴))
113, 10imbi12d 336 . . 3 (𝑔 = 𝐺 → (((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))) ∈ 𝐴)))
12 simp2 1168 . . . 4 ((𝜑𝐹𝐴𝑔𝐴) → 𝐹𝐴)
13 eleq1 2866 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝐴𝐹𝐴))
14133anbi2d 1566 . . . . . 6 (𝑓 = 𝐹 → ((𝜑𝑓𝐴𝑔𝐴) ↔ (𝜑𝐹𝐴𝑔𝐴)))
15 stoweidlem8.2 . . . . . . . . 9 𝑡𝐹
1615nfeq2 2957 . . . . . . . 8 𝑡 𝑓 = 𝐹
17 fveq1 6410 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑡) = (𝐹𝑡))
1817oveq1d 6893 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑡) + (𝑔𝑡)) = ((𝐹𝑡) + (𝑔𝑡)))
1918adantr 473 . . . . . . . 8 ((𝑓 = 𝐹𝑡𝑇) → ((𝑓𝑡) + (𝑔𝑡)) = ((𝐹𝑡) + (𝑔𝑡)))
2016, 19mpteq2da 4936 . . . . . . 7 (𝑓 = 𝐹 → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))))
2120eleq1d 2863 . . . . . 6 (𝑓 = 𝐹 → ((𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴 ↔ (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴))
2214, 21imbi12d 336 . . . . 5 (𝑓 = 𝐹 → (((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴) ↔ ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴)))
23 stoweidlem8.1 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
2422, 23vtoclg 3453 . . . 4 (𝐹𝐴 → ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴))
2512, 24mpcom 38 . . 3 ((𝜑𝐹𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝑔𝑡))) ∈ 𝐴)
2611, 25vtoclg 3453 . 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 1108   = wceq 1653  wcel 2157  wnfc 2928  cmpt 4922  cfv 6101  (class class class)co 6878   + caddc 10227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-iota 6064  df-fv 6109  df-ov 6881
This theorem is referenced by:  stoweidlem20  40980  stoweidlem21  40981  stoweidlem22  40982  stoweidlem23  40983
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