Theorem axcontlem1 28087

Description: Lemma for axcont 28099. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)

Hypothesis

Ref	Expression
axcontlem1.1	⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}

Assertion

Ref	Expression
axcontlem1	⊢ 𝐹 = {⟨𝑦, 𝑠⟩ ∣ (𝑦 ∈ 𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))}

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

Step	Hyp	Ref	Expression
1		axcontlem1.1	. 2 ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}
2		eleq1w 2815	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷))
3	2	adantr 481	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → (𝑥 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷))
4		eleq1w 2815	. . . . . 6 ⊢ (𝑡 = 𝑠 → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
5	4	adantl 482	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
6		fveq1 6877	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥‘𝑖) = (𝑦‘𝑖))
7		oveq2 7401	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (1 − 𝑡) = (1 − 𝑠))
8	7	oveq1d 7408	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((1 − 𝑡) · (𝑍‘𝑖)) = ((1 − 𝑠) · (𝑍‘𝑖)))
9		oveq1 7400	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → (𝑡 · (𝑈‘𝑖)) = (𝑠 · (𝑈‘𝑖)))
10	8, 9	oveq12d 7411	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖))) = (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))))
11	6, 10	eqeqan12d 2745	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖))) ↔ (𝑦‘𝑖) = (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖)))))
12	11	ralbidv 3176	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → (∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖)))))
13		fveq2 6878	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑦‘𝑖) = (𝑦‘𝑗))
14		fveq2 6878	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑍‘𝑖) = (𝑍‘𝑗))
15	14	oveq2d 7409	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((1 − 𝑠) · (𝑍‘𝑖)) = ((1 − 𝑠) · (𝑍‘𝑗)))
16		fveq2 6878	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑈‘𝑖) = (𝑈‘𝑗))
17	16	oveq2d 7409	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑠 · (𝑈‘𝑖)) = (𝑠 · (𝑈‘𝑗)))
18	15, 17	oveq12d 7411	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗))))
19	13, 18	eqeq12d 2747	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝑦‘𝑖) = (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))) ↔ (𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))
20	19	cbvralvw 3233	. . . . . 6 ⊢ (∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))) ↔ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗))))
21	12, 20	bitrdi 286	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → (∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖))) ↔ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))
22	5, 21	anbi12d 631	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → ((𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))) ↔ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗))))))
23	3, 22	anbi12d 631	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → ((𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖))))) ↔ (𝑦 ∈ 𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))))
24	23	cbvopabv 5214	. 2 ⊢ {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))} = {⟨𝑦, 𝑠⟩ ∣ (𝑦 ∈ 𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))}
25	1, 24	eqtri 2759	1 ⊢ 𝐹 = {⟨𝑦, 𝑠⟩ ∣ (𝑦 ∈ 𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))}

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  {copab 5203  ‘cfv 6532  (class class class)co 7393  0cc0 11092  1c1 11093   + caddc 11095   · cmul 11097  +∞cpnf 11227   − cmin 11426  [,)cico 13308  ...cfz 13466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-iota 6484  df-fv 6540  df-ov 7396

This theorem is referenced by:  axcontlem6  28092  axcontlem11  28097