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Theorem axcontlem10 27341

Description: Lemma for axcont 27344. Given a handful of assumptions, derive the conclusion of the final theorem. (Contributed by Scott Fenton, 20-Jun-2013.)

**Hypotheses**
Ref	Expression
axcontlem10.1	⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}
axcontlem10.2	⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}

**Assertion**
Ref	Expression
axcontlem10	⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Distinct variable groups: 𝐴,𝑏,𝑝,𝑥 𝐵,𝑏,𝑝,𝑥,𝑦 𝐷,𝑝,𝑡,𝑥 𝐹,𝑏 𝑖,𝐹,𝑝,𝑡,𝑥 𝑦,𝐹 𝑁,𝑏 𝑖,𝑁,𝑝,𝑡,𝑥 𝑦,𝑁 𝑈,𝑏 𝑈,𝑖,𝑝,𝑡,𝑥 𝑦,𝑈 𝑍,𝑏 𝑖,𝑍,𝑝,𝑡,𝑥 𝑦,𝑍 Allowed substitution hints: 𝐴(𝑦,𝑡,𝑖) 𝐵(𝑡,𝑖) 𝐷(𝑦,𝑖,𝑏)

Proof of Theorem axcontlem10 Dummy variables 𝑘 𝑚 𝑛 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 5980	. . . . 5 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
2		simpll 764	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝑁 ∈ ℕ)
3		simprl1 1217	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝑍 ∈ (𝔼‘𝑁))
4		simplr1 1214	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐴 ⊆ (𝔼‘𝑁))
5		simprl2 1218	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝑈 ∈ 𝐴)
6	4, 5	sseldd 3922	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝑈 ∈ (𝔼‘𝑁))
7		simprr 770	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝑍 ≠ 𝑈)
8		axcontlem10.1	. . . . . . . 8 ⊢ 𝐷 = {𝑝 ∈ (𝔼‘𝑁) ∣ (𝑈 Btwn ⟨𝑍, 𝑝⟩ ∨ 𝑝 Btwn ⟨𝑍, 𝑈⟩)}
9		axcontlem10.2	. . . . . . . 8 ⊢ 𝐹 = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐷 ∧ (𝑡 ∈ (0[,)+∞) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑍‘𝑖)) + (𝑡 · (𝑈‘𝑖)))))}
10	8, 9	axcontlem2 27333	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) → 𝐹:𝐷–1-1-onto→(0[,)+∞))
11	2, 3, 6, 7, 10	syl31anc 1372	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐹:𝐷–1-1-onto→(0[,)+∞))
12		f1ofo 6723	. . . . . 6 ⊢ (𝐹:𝐷–1-1-onto→(0[,)+∞) → 𝐹:𝐷–onto→(0[,)+∞))
13		forn 6691	. . . . . 6 ⊢ (𝐹:𝐷–onto→(0[,)+∞) → ran 𝐹 = (0[,)+∞))
14	11, 12, 13	3syl 18	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ran 𝐹 = (0[,)+∞))
15	1, 14	sseqtrid 3973	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝐹 “ 𝐴) ⊆ (0[,)+∞))
16		rge0ssre 13188	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ
17	15, 16	sstrdi 3933	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝐹 “ 𝐴) ⊆ ℝ)
18		imassrn 5980	. . . . 5 ⊢ (𝐹 “ 𝐵) ⊆ ran 𝐹
19	18, 14	sseqtrid 3973	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝐹 “ 𝐵) ⊆ (0[,)+∞))
20	19, 16	sstrdi 3933	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝐹 “ 𝐵) ⊆ ℝ)
21	8, 9	axcontlem9 27340	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)𝑚 ≤ 𝑛)
22		dedekindle 11139	. . 3 ⊢ (((𝐹 “ 𝐴) ⊆ ℝ ∧ (𝐹 “ 𝐵) ⊆ ℝ ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)𝑚 ≤ 𝑛) → ∃𝑘 ∈ ℝ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛))
23	17, 20, 21, 22	syl3anc 1370	. 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∃𝑘 ∈ ℝ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛))
24		simpr 485	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ 𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
25		simprl3 1219	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐵 ≠ ∅)
26	25	ad2antrr 723	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ 𝑘 ∈ ℝ) → 𝐵 ≠ ∅)
27		n0 4280	. . . . . . . . . 10 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐵)
28	26, 27	sylib 217	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ 𝑘 ∈ ℝ) → ∃𝑏 𝑏 ∈ 𝐵)
29		0red 10978	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → 0 ∈ ℝ)
30		f1of 6716	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝐷–1-1-onto→(0[,)+∞) → 𝐹:𝐷⟶(0[,)+∞))
31	11, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐹:𝐷⟶(0[,)+∞))
32	8	axcontlem4 27335	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐴 ⊆ 𝐷)
33	32, 5	sseldd 3922	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝑈 ∈ 𝐷)
34	31, 33	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝐹‘𝑈) ∈ (0[,)+∞))
35	16, 34	sselid 3919	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝐹‘𝑈) ∈ ℝ)
36	35	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑈) ∈ ℝ)
37		simprl 768	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → 𝑘 ∈ ℝ)
38		elrege0 13186	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑈) ∈ (0[,)+∞) ↔ ((𝐹‘𝑈) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑈)))
39	38	simprbi 497	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑈) ∈ (0[,)+∞) → 0 ≤ (𝐹‘𝑈))
40	34, 39	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 0 ≤ (𝐹‘𝑈))
41	40	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → 0 ≤ (𝐹‘𝑈))
42		f1of1 6715	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹:𝐷–1-1-onto→(0[,)+∞) → 𝐹:𝐷–1-1→(0[,)+∞))
43	11, 42	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐹:𝐷–1-1→(0[,)+∞))
44		f1elima 7136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:𝐷–1-1→(0[,)+∞) ∧ 𝑈 ∈ 𝐷 ∧ 𝐴 ⊆ 𝐷) → ((𝐹‘𝑈) ∈ (𝐹 “ 𝐴) ↔ 𝑈 ∈ 𝐴))
45	43, 33, 32, 44	syl3anc 1370	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ((𝐹‘𝑈) ∈ (𝐹 “ 𝐴) ↔ 𝑈 ∈ 𝐴))
46	5, 45	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝐹‘𝑈) ∈ (𝐹 “ 𝐴))
47	46	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑈) ∈ (𝐹 “ 𝐴))
48		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
49	43	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ 𝑏 ∈ 𝐵) → 𝐹:𝐷–1-1→(0[,)+∞))
50		simpl1 1190	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈) → 𝑍 ∈ (𝔼‘𝑁))
51		simpl2 1191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈) → 𝑈 ∈ 𝐴)
52		simpr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈) → 𝑍 ≠ 𝑈)
53	50, 51, 52	3jca 1127	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈) → (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈))
54	8	axcontlem3 27334	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ (𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝑍 ≠ 𝑈)) → 𝐵 ⊆ 𝐷)
55	53, 54	sylan2 593	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐵 ⊆ 𝐷)
56	55	sselda 3921	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐷)
57	55	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ 𝑏 ∈ 𝐵) → 𝐵 ⊆ 𝐷)
58		f1elima 7136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹:𝐷–1-1→(0[,)+∞) ∧ 𝑏 ∈ 𝐷 ∧ 𝐵 ⊆ 𝐷) → ((𝐹‘𝑏) ∈ (𝐹 “ 𝐵) ↔ 𝑏 ∈ 𝐵))
59	49, 56, 57, 58	syl3anc 1370	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ 𝑏 ∈ 𝐵) → ((𝐹‘𝑏) ∈ (𝐹 “ 𝐵) ↔ 𝑏 ∈ 𝐵))
60	48, 59	mpbird 256	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) ∈ (𝐹 “ 𝐵))
61	60	adantrl 713	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) ∈ (𝐹 “ 𝐵))
62	47, 61	jca 512	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → ((𝐹‘𝑈) ∈ (𝐹 “ 𝐴) ∧ (𝐹‘𝑏) ∈ (𝐹 “ 𝐵)))
63		breq1 5077	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝐹‘𝑈) → (𝑚 ≤ 𝑘 ↔ (𝐹‘𝑈) ≤ 𝑘))
64	63	anbi1d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐹‘𝑈) → ((𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ ((𝐹‘𝑈) ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)))
65		breq2 5078	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝐹‘𝑏) → (𝑘 ≤ 𝑛 ↔ 𝑘 ≤ (𝐹‘𝑏)))
66	65	anbi2d 629	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝐹‘𝑏) → (((𝐹‘𝑈) ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ ((𝐹‘𝑈) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑏))))
67	64, 66	rspc2va 3571	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹‘𝑈) ∈ (𝐹 “ 𝐴) ∧ (𝐹‘𝑏) ∈ (𝐹 “ 𝐵)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) → ((𝐹‘𝑈) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑏)))
68	62, 67	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) → ((𝐹‘𝑈) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑏)))
69	68	an32s 649	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → ((𝐹‘𝑈) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑏)))
70	69	simpld 495	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑈) ≤ 𝑘)
71	29, 36, 37, 41, 70	letrd 11132	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ (𝑘 ∈ ℝ ∧ 𝑏 ∈ 𝐵)) → 0 ≤ 𝑘)
72	71	expr 457	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ 𝑘 ∈ ℝ) → (𝑏 ∈ 𝐵 → 0 ≤ 𝑘))
73	72	exlimdv 1936	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ 𝑘 ∈ ℝ) → (∃𝑏 𝑏 ∈ 𝐵 → 0 ≤ 𝑘))
74	28, 73	mpd 15	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ 𝑘 ∈ ℝ) → 0 ≤ 𝑘)
75		elrege0 13186	. . . . . . . 8 ⊢ (𝑘 ∈ (0[,)+∞) ↔ (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘))
76	24, 74, 75	sylanbrc 583	. . . . . . 7 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) ∧ 𝑘 ∈ ℝ) → 𝑘 ∈ (0[,)+∞))
77	76	ex 413	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) → (𝑘 ∈ ℝ → 𝑘 ∈ (0[,)+∞)))
78	8	ssrab3 4015	. . . . . . . . 9 ⊢ 𝐷 ⊆ (𝔼‘𝑁)
79		simpr 485	. . . . . . . . . 10 ⊢ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) → 𝑘 ∈ (0[,)+∞))
80		f1ocnvdm 7157	. . . . . . . . . 10 ⊢ ((𝐹:𝐷–1-1-onto→(0[,)+∞) ∧ 𝑘 ∈ (0[,)+∞)) → (^◡𝐹‘𝑘) ∈ 𝐷)
81	11, 79, 80	syl2an 596	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞))) → (^◡𝐹‘𝑘) ∈ 𝐷)
82	78, 81	sselid 3919	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞))) → (^◡𝐹‘𝑘) ∈ (𝔼‘𝑁))
83	2, 3, 6	3jca 1127	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)))
84	83, 7	jca 512	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈))
85	84	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → ((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈))
86	32	sselda 3921	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐷)
87	86	adantrr 714	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵)) → 𝑞 ∈ 𝐷)
88	87	adantrl 713	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → 𝑞 ∈ 𝐷)
89		simplr 766	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵)) → 𝑘 ∈ (0[,)+∞))
90	11, 89, 80	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → (^◡𝐹‘𝑘) ∈ 𝐷)
91	55	sselda 3921	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ 𝑟 ∈ 𝐵) → 𝑟 ∈ 𝐷)
92	91	adantrl 713	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵)) → 𝑟 ∈ 𝐷)
93	92	adantrl 713	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → 𝑟 ∈ 𝐷)
94	88, 90, 93	3jca 1127	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → (𝑞 ∈ 𝐷 ∧ (^◡𝐹‘𝑘) ∈ 𝐷 ∧ 𝑟 ∈ 𝐷))
95	85, 94	jca 512	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → (((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ (𝑞 ∈ 𝐷 ∧ (^◡𝐹‘𝑘) ∈ 𝐷 ∧ 𝑟 ∈ 𝐷)))
96		f1ofun 6718	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝐷–1-1-onto→(0[,)+∞) → Fun 𝐹)
97	11, 96	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → Fun 𝐹)
98		fdm 6609	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹:𝐷⟶(0[,)+∞) → dom 𝐹 = 𝐷)
99	11, 30, 98	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → dom 𝐹 = 𝐷)
100	32, 99	sseqtrrd 3962	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐴 ⊆ dom 𝐹)
101		funfvima2 7107	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑞 ∈ 𝐴 → (𝐹‘𝑞) ∈ (𝐹 “ 𝐴)))
102	97, 100, 101	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝑞 ∈ 𝐴 → (𝐹‘𝑞) ∈ (𝐹 “ 𝐴)))
103	55, 99	sseqtrrd 3962	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → 𝐵 ⊆ dom 𝐹)
104		funfvima2 7107	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝑟 ∈ 𝐵 → (𝐹‘𝑟) ∈ (𝐹 “ 𝐵)))
105	97, 103, 104	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝑟 ∈ 𝐵 → (𝐹‘𝑟) ∈ (𝐹 “ 𝐵)))
106	102, 105	anim12d 609	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ((𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵) → ((𝐹‘𝑞) ∈ (𝐹 “ 𝐴) ∧ (𝐹‘𝑟) ∈ (𝐹 “ 𝐵))))
107	106	imp 407	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵)) → ((𝐹‘𝑞) ∈ (𝐹 “ 𝐴) ∧ (𝐹‘𝑟) ∈ (𝐹 “ 𝐵)))
108	107	adantrl 713	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → ((𝐹‘𝑞) ∈ (𝐹 “ 𝐴) ∧ (𝐹‘𝑟) ∈ (𝐹 “ 𝐵)))
109		simprll 776	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛))
110		breq1 5077	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐹‘𝑞) → (𝑚 ≤ 𝑘 ↔ (𝐹‘𝑞) ≤ 𝑘))
111	110	anbi1d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐹‘𝑞) → ((𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ ((𝐹‘𝑞) ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)))
112		breq2 5078	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝐹‘𝑟) → (𝑘 ≤ 𝑛 ↔ 𝑘 ≤ (𝐹‘𝑟)))
113	112	anbi2d 629	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝐹‘𝑟) → (((𝐹‘𝑞) ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ ((𝐹‘𝑞) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑟))))
114	111, 113	rspc2v 3570	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑞) ∈ (𝐹 “ 𝐴) ∧ (𝐹‘𝑟) ∈ (𝐹 “ 𝐵)) → (∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) → ((𝐹‘𝑞) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑟))))
115	108, 109, 114	sylc 65	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → ((𝐹‘𝑞) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑟)))
116		f1ocnvfv2 7149	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐷–1-1-onto→(0[,)+∞) ∧ 𝑘 ∈ (0[,)+∞)) → (𝐹‘(^◡𝐹‘𝑘)) = 𝑘)
117	11, 89, 116	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → (𝐹‘(^◡𝐹‘𝑘)) = 𝑘)
118	117	breq2d 5086	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → ((𝐹‘𝑞) ≤ (𝐹‘(^◡𝐹‘𝑘)) ↔ (𝐹‘𝑞) ≤ 𝑘))
119	117	breq1d 5084	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → ((𝐹‘(^◡𝐹‘𝑘)) ≤ (𝐹‘𝑟) ↔ 𝑘 ≤ (𝐹‘𝑟)))
120	118, 119	anbi12d 631	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → (((𝐹‘𝑞) ≤ (𝐹‘(^◡𝐹‘𝑘)) ∧ (𝐹‘(^◡𝐹‘𝑘)) ≤ (𝐹‘𝑟)) ↔ ((𝐹‘𝑞) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑟))))
121	115, 120	mpbird 256	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → ((𝐹‘𝑞) ≤ (𝐹‘(^◡𝐹‘𝑘)) ∧ (𝐹‘(^◡𝐹‘𝑘)) ≤ (𝐹‘𝑟)))
122	8, 9	axcontlem8 27339	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ (𝔼‘𝑁)) ∧ 𝑍 ≠ 𝑈) ∧ (𝑞 ∈ 𝐷 ∧ (^◡𝐹‘𝑘) ∈ 𝐷 ∧ 𝑟 ∈ 𝐷)) → (((𝐹‘𝑞) ≤ (𝐹‘(^◡𝐹‘𝑘)) ∧ (𝐹‘(^◡𝐹‘𝑘)) ≤ (𝐹‘𝑟)) → (^◡𝐹‘𝑘) Btwn ⟨𝑞, 𝑟⟩))
123	95, 121, 122	sylc 65	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵))) → (^◡𝐹‘𝑘) Btwn ⟨𝑞, 𝑟⟩)
124	123	anassrs 468	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞))) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐵)) → (^◡𝐹‘𝑘) Btwn ⟨𝑞, 𝑟⟩)
125	124	ralrimivva 3123	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞))) → ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐵 (^◡𝐹‘𝑘) Btwn ⟨𝑞, 𝑟⟩)
126		opeq1 4804	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑥 → ⟨𝑞, 𝑟⟩ = ⟨𝑥, 𝑟⟩)
127	126	breq2d 5086	. . . . . . . . . 10 ⊢ (𝑞 = 𝑥 → ((^◡𝐹‘𝑘) Btwn ⟨𝑞, 𝑟⟩ ↔ (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑟⟩))
128		opeq2 4805	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑦 → ⟨𝑥, 𝑟⟩ = ⟨𝑥, 𝑦⟩)
129	128	breq2d 5086	. . . . . . . . . 10 ⊢ (𝑟 = 𝑦 → ((^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑟⟩ ↔ (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑦⟩))
130	127, 129	cbvral2vw 3396	. . . . . . . . 9 ⊢ (∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐵 (^◡𝐹‘𝑘) Btwn ⟨𝑞, 𝑟⟩ ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑦⟩)
131	125, 130	sylib 217	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞))) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑦⟩)
132		breq1 5077	. . . . . . . . . 10 ⊢ (𝑏 = (^◡𝐹‘𝑘) → (𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑦⟩))
133	132	2ralbidv 3129	. . . . . . . . 9 ⊢ (𝑏 = (^◡𝐹‘𝑘) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑦⟩))
134	133	rspcev 3561	. . . . . . . 8 ⊢ (((^◡𝐹‘𝑘) ∈ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑦⟩) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)
135	82, 131, 134	syl2anc 584	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ (∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞))) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)
136	135	expr 457	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) → (𝑘 ∈ (0[,)+∞) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩))
137	77, 136	syld 47	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) ∧ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)) → (𝑘 ∈ ℝ → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩))
138	137	ex 413	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) → (𝑘 ∈ ℝ → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)))
139	138	com23 86	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (𝑘 ∈ ℝ → (∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)))
140	139	rexlimdv 3212	. 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → (∃𝑘 ∈ ℝ ∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩))
141	23, 140	mpd 15	1 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ⊆ (𝔼‘𝑁) ∧ 𝐵 ⊆ (𝔼‘𝑁) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 Btwn ⟨𝑍, 𝑦⟩)) ∧ ((𝑍 ∈ (𝔼‘𝑁) ∧ 𝑈 ∈ 𝐴 ∧ 𝐵 ≠ ∅) ∧ 𝑍 ≠ 𝑈)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 ⊆ wss 3887 ∅c0 4256 ⟨cop 4567 class class class wbr 5074 {copab 5136 ^◡ccnv 5588 dom cdm 5589 ran crn 5590 “ cima 5592 Fun wfun 6427 ⟶wf 6429 –1-1→wf1 6430 –onto→wfo 6431 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 +∞cpnf 11006 ≤ cle 11010 − cmin 11205 ℕcn 11973 [,)cico 13081 ...cfz 13239 𝔼cee 27256 Btwn cbtwn 27257

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-z 12320 df-uz 12583 df-ico 13085 df-icc 13086 df-fz 13240 df-ee 27259 df-btwn 27260

This theorem is referenced by: axcontlem11 27342

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