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Theorem axcontlem1 28994
Description: Lemma for axcont 29006. 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 2822 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐷𝑦𝐷))
32adantr 480 . . . 4 ((𝑥 = 𝑦𝑡 = 𝑠) → (𝑥𝐷𝑦𝐷))
4 eleq1w 2822 . . . . . 6 (𝑡 = 𝑠 → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
54adantl 481 . . . . 5 ((𝑥 = 𝑦𝑡 = 𝑠) → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
6 fveq1 6906 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝑖) = (𝑦𝑖))
7 oveq2 7439 . . . . . . . . . 10 (𝑡 = 𝑠 → (1 − 𝑡) = (1 − 𝑠))
87oveq1d 7446 . . . . . . . . 9 (𝑡 = 𝑠 → ((1 − 𝑡) · (𝑍𝑖)) = ((1 − 𝑠) · (𝑍𝑖)))
9 oveq1 7438 . . . . . . . . 9 (𝑡 = 𝑠 → (𝑡 · (𝑈𝑖)) = (𝑠 · (𝑈𝑖)))
108, 9oveq12d 7449 . . . . . . . 8 (𝑡 = 𝑠 → (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))))
116, 10eqeqan12d 2749 . . . . . . 7 ((𝑥 = 𝑦𝑡 = 𝑠) → ((𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ (𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖)))))
1211ralbidv 3176 . . . . . 6 ((𝑥 = 𝑦𝑡 = 𝑠) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑡) · (𝑍𝑖)) + (𝑡 · (𝑈𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖)))))
13 fveq2 6907 . . . . . . . 8 (𝑖 = 𝑗 → (𝑦𝑖) = (𝑦𝑗))
14 fveq2 6907 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑍𝑖) = (𝑍𝑗))
1514oveq2d 7447 . . . . . . . . 9 (𝑖 = 𝑗 → ((1 − 𝑠) · (𝑍𝑖)) = ((1 − 𝑠) · (𝑍𝑗)))
16 fveq2 6907 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑈𝑖) = (𝑈𝑗))
1716oveq2d 7447 . . . . . . . . 9 (𝑖 = 𝑗 → (𝑠 · (𝑈𝑖)) = (𝑠 · (𝑈𝑗)))
1815, 17oveq12d 7449 . . . . . . . 8 (𝑖 = 𝑗 → (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗))))
1913, 18eqeq12d 2751 . . . . . . 7 (𝑖 = 𝑗 → ((𝑦𝑖) = (((1 − 𝑠) · (𝑍𝑖)) + (𝑠 · (𝑈𝑖))) ↔ (𝑦𝑗) = (((1 − 𝑠) · (𝑍𝑗)) + (𝑠 · (𝑈𝑗)))))
2019cbvralvw 3235 . . . . . 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 5221 . 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 2763 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 1537  wcel 2106  wral 3059  {copab 5210  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  +∞cpnf 11290  cmin 11490  [,)cico 13386  ...cfz 13544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  axcontlem6  28999  axcontlem11  29004
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