Theorem axcontlem1 29542

Description: Lemma for axcont 29554. 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 2844	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷))
3	2	adantr 486	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → (𝑥 ∈ 𝐷 ↔ 𝑦 ∈ 𝐷))
4		eleq1w 2844	. . . . . 6 ⊢ (𝑡 = 𝑠 → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
5	4	adantl 487	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑠) → (𝑡 ∈ (0[,)+∞) ↔ 𝑠 ∈ (0[,)+∞)))
6		fveq1 6884	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥‘𝑖) = (𝑦‘𝑖))
7		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (1 − 𝑡) = (1 − 𝑠))
8	7	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((1 − 𝑡) · (𝑍‘𝑖)) = ((1 − 𝑠) · (𝑍‘𝑖)))
9		oveq1 7427	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → (𝑡 · (𝑈‘𝑖)) = (𝑠 · (𝑈‘𝑖)))
10	8, 9	oveq12d 7438	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖))) = (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))))
11	6, 10	eqeqan12d 2775	. . . . . . 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 3186	. . . . . 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 6885	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (𝑦‘𝑖) = (𝑦‘𝑗))
14		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑍‘𝑖) = (𝑍‘𝑗))
15	14	oveq2d 7436	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((1 − 𝑠) · (𝑍‘𝑖)) = ((1 − 𝑠) · (𝑍‘𝑗)))
16		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑈‘𝑖) = (𝑈‘𝑗))
17	16	oveq2d 7436	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑠 · (𝑈‘𝑖)) = (𝑠 · (𝑈‘𝑗)))
18	15, 17	oveq12d 7438	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗))))
19	13, 18	eqeq12d 2777	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝑦‘𝑖) = (((1 − 𝑠) · (𝑍‘𝑖)) + (𝑠 · (𝑈‘𝑖))) ↔ (𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))
20	19	cbvralvw 3241	. . . . . 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 290	. . . . 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 644	. . . 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 644	. . 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 5178	. 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 2784	1 ⊢ 𝐹 = {⟨𝑦, 𝑠⟩ ∣ (𝑦 ∈ 𝐷 ∧ (𝑠 ∈ (0[,)+∞) ∧ ∀𝑗 ∈ (1...𝑁)(𝑦‘𝑗) = (((1 − 𝑠) · (𝑍‘𝑗)) + (𝑠 · (𝑈‘𝑗)))))}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {copab 5167  ‘cfv 6538  (class class class)co 7420  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205  +∞cpnf 11340   − cmin 11541  [,)cico 13478  ...cfz 13639

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by:  axcontlem6  29547  axcontlem11  29552