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Theorem axcontlem1 28980
Description: Lemma for axcont 28992. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)
Hypothesis
Ref Expression
axcontlem1.1 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}
Assertion
Ref Expression
axcontlem1 𝐹 = {⟨𝑦, 𝑠⟩ ∣ (𝑦𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))}
Distinct variable groups:   𝐷,𝑠,𝑡,𝑥,𝑦   𝑖,𝑗,𝑠,𝑡,𝑥,𝑦,𝑁   𝑈,𝑖,𝑗,𝑠,𝑡,𝑥,𝑦   𝑖,𝑍,𝑗,𝑠,𝑡,𝑥,𝑦
Allowed substitution hints:   𝐷(𝑖,𝑗)   𝐹(𝑥,𝑦,𝑡,𝑖,𝑗,𝑠)

Proof of Theorem axcontlem1
StepHypRef Expression
1 axcontlem1.1 . 2 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))}
2 eleq1w 2823 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐷𝑦𝐷))
32adantr 480 . . . 4 ((𝑥 = 𝑦𝑡 = 𝑠) → (𝑥𝐷𝑦𝐷))
4 eleq1w 2823 . . . . . 6 (𝑡 = 𝑠 → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
54adantl 481 . . . . 5 ((𝑥 = 𝑦𝑡 = 𝑠) → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
6 fveq1 6904 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑖) = (𝑦𝑖))
7 oveq2 7440 . . . . . . . . . 10 (𝑡 = 𝑠 → (1 − 𝑡) = (1 − 𝑠))
87oveq1d 7447 . . . . . . . . 9 (𝑡 = 𝑠 → ((1 − 𝑡) · (𝑍𝑖)) = ((1 − 𝑠) · (𝑍𝑖)))
9 oveq1 7439 . . . . . . . . 9 (𝑡 = 𝑠 → (𝑡 · (𝑈𝑖)) = (𝑠 · (𝑈𝑖)))
108, 9oveq12d 7450 . . . . . . . 8 (𝑡 = 𝑠 → (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))))
116, 10eqeqan12d 2750 . . . . . . 7 ((𝑥 = 𝑦𝑡 = 𝑠) → ((𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ (𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖)))))
1211ralbidv 3177 . . . . . 6 ((𝑥 = 𝑦𝑡 = 𝑠) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖)))))
13 fveq2 6905 . . . . . . . 8 (𝑖 = 𝑗 → (𝑦𝑖) = (𝑦𝑗))
14 fveq2 6905 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑍𝑖) = (𝑍𝑗))
1514oveq2d 7448 . . . . . . . . 9 (𝑖 = 𝑗 → ((1 − 𝑠) · (𝑍𝑖)) = ((1 − 𝑠) · (𝑍𝑗)))
16 fveq2 6905 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑈𝑖) = (𝑈𝑗))
1716oveq2d 7448 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑠 · (𝑈𝑖)) = (𝑠 · (𝑈𝑗)))
1815, 17oveq12d 7450 . . . . . . . 8 (𝑖 = 𝑗 → (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗))))
1913, 18eqeq12d 2752 . . . . . . 7 (𝑖 = 𝑗 → ((𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) ↔ (𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))
2019cbvralvw 3236 . . . . . 6 (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) ↔ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗))))
2112, 20bitrdi 287 . . . . 5 ((𝑥 = 𝑦𝑡 = 𝑠) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))
225, 21anbi12d 632 . . . 4 ((𝑥 = 𝑦𝑡 = 𝑠) → ((𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))) ↔ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗))))))
233, 22anbi12d 632 . . 3 ((𝑥 = 𝑦𝑡 = 𝑠) → ((𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))))) ↔ (𝑦𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))))
2423cbvopabv 5215 . 2 {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖)))))} = {⟨𝑦, 𝑠⟩ ∣ (𝑦𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))}
251, 24eqtri 2764 1 𝐹 = {⟨𝑦, 𝑠⟩ ∣ (𝑦𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  {copab 5204  cfv 6560  (class class class)co 7432  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161  +∞cpnf 11293  cmin 11493  [,)cico 13390  ...cfz 13548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  axcontlem6  28985  axcontlem11  28990
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