Theorem axcontlem1 29162

Description: Lemma for axcont 29174. 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 2845	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷))
3	2	adantr 484	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → (𝑥 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷))
4		eleq1w 2845	. . . . . 6 ⊢ (𝑡 = 𝑠 → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
5	4	adantl 485	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
6		fveq1 6866	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥‘𝑖) = (𝑦‘𝑖))
7		oveq2 7404	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (1 − 𝑡) = (1 − 𝑠))
8	7	oveq1d 7411	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((1 − 𝑡) · (𝑍‘𝑖)) = ((1 − 𝑠) · (𝑍‘𝑖)))
9		oveq1 7403	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → (𝑡 · (𝑈‘𝑖)) = (𝑠 · (𝑈‘𝑖)))
10	8, 9	oveq12d 7414	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖))) = (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))))
11	6, 10	eqeqan12d 2776	. . . . . . 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 3185	. . . . . 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 6867	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑦‘𝑖) = (𝑦‘𝑗))
14		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑍‘𝑖) = (𝑍‘𝑗))
15	14	oveq2d 7412	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((1 − 𝑠) · (𝑍‘𝑖)) = ((1 − 𝑠) · (𝑍‘𝑗)))
16		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑈‘𝑖) = (𝑈‘𝑗))
17	16	oveq2d 7412	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑠 · (𝑈‘𝑖)) = (𝑠 · (𝑈‘𝑗)))
18	15, 17	oveq12d 7414	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗))))
19	13, 18	eqeq12d 2778	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝑦‘𝑖) = (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))) ↔ (𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))
20	19	cbvralvw 3240	. . . . . 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 289	. . . . 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 641	. . . 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 641	. . 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 5173	. 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 2785	1 ⊢ 𝐹 = {⟨𝑦, 𝑠⟩ ∣ (𝑦 ∈ 𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))}

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  {copab 5162  ‘cfv 6521  (class class class)co 7396  0cc0 11073  1c1 11074   + caddc 11076   · cmul 11078  +∞cpnf 11213   − cmin 11414  [,)cico 13351  ...cfz 13512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-iota 6477  df-fv 6529  df-ov 7399

This theorem is referenced by:  axcontlem6  29167  axcontlem11  29172