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Theorem axcontlem1 25644
 Description: Lemma for axcont 25656. 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 eleq1 2676 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐷𝑦𝐷))
32adantr 480 . . . 4 ((𝑥 = 𝑦𝑡 = 𝑠) → (𝑥𝐷𝑦𝐷))
4 eleq1 2676 . . . . . 6 (𝑡 = 𝑠 → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
54adantl 481 . . . . 5 ((𝑥 = 𝑦𝑡 = 𝑠) → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
6 fveq1 6102 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑖) = (𝑦𝑖))
7 oveq2 6557 . . . . . . . . . 10 (𝑡 = 𝑠 → (1 − 𝑡) = (1 − 𝑠))
87oveq1d 6564 . . . . . . . . 9 (𝑡 = 𝑠 → ((1 − 𝑡) · (𝑍𝑖)) = ((1 − 𝑠) · (𝑍𝑖)))
9 oveq1 6556 . . . . . . . . 9 (𝑡 = 𝑠 → (𝑡 · (𝑈𝑖)) = (𝑠 · (𝑈𝑖)))
108, 9oveq12d 6567 . . . . . . . 8 (𝑡 = 𝑠 → (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))))
116, 10eqeqan12d 2626 . . . . . . 7 ((𝑥 = 𝑦𝑡 = 𝑠) → ((𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ (𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖)))))
1211ralbidv 2969 . . . . . 6 ((𝑥 = 𝑦𝑡 = 𝑠) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖)))))
13 fveq2 6103 . . . . . . . 8 (𝑖 = 𝑗 → (𝑦𝑖) = (𝑦𝑗))
14 fveq2 6103 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑍𝑖) = (𝑍𝑗))
1514oveq2d 6565 . . . . . . . . 9 (𝑖 = 𝑗 → ((1 − 𝑠) · (𝑍𝑖)) = ((1 − 𝑠) · (𝑍𝑗)))
16 fveq2 6103 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑈𝑖) = (𝑈𝑗))
1716oveq2d 6565 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑠 · (𝑈𝑖)) = (𝑠 · (𝑈𝑗)))
1815, 17oveq12d 6567 . . . . . . . 8 (𝑖 = 𝑗 → (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗))))
1913, 18eqeq12d 2625 . . . . . . 7 (𝑖 = 𝑗 → ((𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) ↔ (𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))
2019cbvralv 3147 . . . . . 6 (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) ↔ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗))))
2112, 20syl6bb 275 . . . . 5 ((𝑥 = 𝑦𝑡 = 𝑠) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ ∀𝑗 ∈ (1...𝑁)(𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))
225, 21anbi12d 743 . . . 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 743 . . 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 4654 . 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 2632 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {copab 4642  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950   − cmin 10145  [,)cico 12048  ...cfz 12197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  axcontlem6  25649  axcontlem11  25654
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