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Theorem axcontlem1 26910
Description: Lemma for axcont 26922. 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 2815 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐷𝑦𝐷))
32adantr 484 . . . 4 ((𝑥 = 𝑦𝑡 = 𝑠) → (𝑥𝐷𝑦𝐷))
4 eleq1w 2815 . . . . . 6 (𝑡 = 𝑠 → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
54adantl 485 . . . . 5 ((𝑥 = 𝑦𝑡 = 𝑠) → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
6 fveq1 6673 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑖) = (𝑦𝑖))
7 oveq2 7178 . . . . . . . . . 10 (𝑡 = 𝑠 → (1 − 𝑡) = (1 − 𝑠))
87oveq1d 7185 . . . . . . . . 9 (𝑡 = 𝑠 → ((1 − 𝑡) · (𝑍𝑖)) = ((1 − 𝑠) · (𝑍𝑖)))
9 oveq1 7177 . . . . . . . . 9 (𝑡 = 𝑠 → (𝑡 · (𝑈𝑖)) = (𝑠 · (𝑈𝑖)))
108, 9oveq12d 7188 . . . . . . . 8 (𝑡 = 𝑠 → (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))))
116, 10eqeqan12d 2755 . . . . . . 7 ((𝑥 = 𝑦𝑡 = 𝑠) → ((𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ (𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖)))))
1211ralbidv 3109 . . . . . 6 ((𝑥 = 𝑦𝑡 = 𝑠) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖)))))
13 fveq2 6674 . . . . . . . 8 (𝑖 = 𝑗 → (𝑦𝑖) = (𝑦𝑗))
14 fveq2 6674 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑍𝑖) = (𝑍𝑗))
1514oveq2d 7186 . . . . . . . . 9 (𝑖 = 𝑗 → ((1 − 𝑠) · (𝑍𝑖)) = ((1 − 𝑠) · (𝑍𝑗)))
16 fveq2 6674 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑈𝑖) = (𝑈𝑗))
1716oveq2d 7186 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑠 · (𝑈𝑖)) = (𝑠 · (𝑈𝑗)))
1815, 17oveq12d 7188 . . . . . . . 8 (𝑖 = 𝑗 → (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗))))
1913, 18eqeq12d 2754 . . . . . . 7 (𝑖 = 𝑗 → ((𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) ↔ (𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))
2019cbvralvw 3349 . . . . . 6 (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) ↔ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗))))
2112, 20bitrdi 290 . . . . 5 ((𝑥 = 𝑦𝑡 = 𝑠) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))
225, 21anbi12d 634 . . . 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 634 . . 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 5102 . 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 2761 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 209  wa 399   = wceq 1542  wcel 2114  wral 3053  {copab 5092  cfv 6339  (class class class)co 7170  0cc0 10615  1c1 10616   + caddc 10618   · cmul 10620  +∞cpnf 10750  cmin 10948  [,)cico 12823  ...cfz 12981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-iota 6297  df-fv 6347  df-ov 7173
This theorem is referenced by:  axcontlem6  26915  axcontlem11  26920
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