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Theorem axcontlem10 28666

Description: Lemma for axcont 28669. 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 6070	. . . . 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 3983	. . . . . . 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 28658	. . . . . . 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 6840	. . . . . 6 ⊢ (𝐹:𝐷–1-1-onto→(0[,)+∞) → 𝐹:𝐷–onto→(0[,)+∞))
13		forn 6808	. . . . . 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 4034	. . . 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 13440	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ
17	15, 16	sstrdi 3994	. . 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 6070	. . . . 5 ⊢ (𝐹 “ 𝐵) ⊆ ran 𝐹
19	18, 14	sseqtrid 4034	. . . 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 3994	. . 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 28665	. . 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 11385	. . 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 484	. . . . . . . 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 4346	. . . . . . . . . 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 11224	. . . . . . . . . . . 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 6833	. . . . . . . . . . . . . . . 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 28660	. . . . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . . . 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	ffvelcdmd 7087	. . . . . . . . . . . . . 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 3980	. . . . . . . . . . . . 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 13438	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑈) ∈ (0[,)+∞) ↔ ((𝐹‘𝑈) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑈)))
39	38	simprbi 496	. . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . . . . . 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 7265	. . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . 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 28659	. . . . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 7265	. . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . 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 5151	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝐹‘𝑈) → (𝑚 ≤ 𝑘 ↔ (𝐹‘𝑈) ≤ 𝑘))
64	63	anbi1d 629	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐹‘𝑈) → ((𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ ((𝐹‘𝑈) ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)))
65		breq2 5152	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝐹‘𝑏) → (𝑘 ≤ 𝑛 ↔ 𝑘 ≤ (𝐹‘𝑏)))
66	65	anbi2d 628	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝐹‘𝑏) → (((𝐹‘𝑈) ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ ((𝐹‘𝑈) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑏))))
67	64, 66	rspc2va 3623	. . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . 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 494	. . . . . . . . . . . 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 11378	. . . . . . . . . . 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 456	. . . . . . . . . 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 1935	. . . . . . . . 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 13438	. . . . . . . 8 ⊢ (𝑘 ∈ (0[,)+∞) ↔ (𝑘 ∈ ℝ ∧ 0 ≤ 𝑘))
76	24, 74, 75	sylanbrc 582	. . . . . . 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 412	. . . . . 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 4080	. . . . . . . . 9 ⊢ 𝐷 ⊆ (𝔼‘𝑁)
79		simpr 484	. . . . . . . . . 10 ⊢ ((∀𝑚 ∈ (𝐹 “ 𝐴)∀𝑛 ∈ (𝐹 “ 𝐵)(𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ∧ 𝑘 ∈ (0[,)+∞)) → 𝑘 ∈ (0[,)+∞))
80		f1ocnvdm 7286	. . . . . . . . . 10 ⊢ ((𝐹:𝐷–1-1-onto→(0[,)+∞) ∧ 𝑘 ∈ (0[,)+∞)) → (^◡𝐹‘𝑘) ∈ 𝐷)
81	11, 79, 80	syl2an 595	. . . . . . . . 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 3980	. . . . . . . 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 511	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 6835	. . . . . . . . . . . . . . . . . . 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 6726	. . . . . . . . . . . . . . . . . . . 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 4023	. . . . . . . . . . . . . . . . . 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 7235	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑞 ∈ 𝐴 → (𝐹‘𝑞) ∈ (𝐹 “ 𝐴)))
102	97, 100, 101	syl2anc 583	. . . . . . . . . . . . . . . . 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 4023	. . . . . . . . . . . . . . . . . 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 7235	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝑟 ∈ 𝐵 → (𝐹‘𝑟) ∈ (𝐹 “ 𝐵)))
105	97, 103, 104	syl2anc 583	. . . . . . . . . . . . . . . . 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 608	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . 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 5151	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐹‘𝑞) → (𝑚 ≤ 𝑘 ↔ (𝐹‘𝑞) ≤ 𝑘))
111	110	anbi1d 629	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐹‘𝑞) → ((𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ ((𝐹‘𝑞) ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)))
112		breq2 5152	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝐹‘𝑟) → (𝑘 ≤ 𝑛 ↔ 𝑘 ≤ (𝐹‘𝑟)))
113	112	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝐹‘𝑟) → (((𝐹‘𝑞) ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ ((𝐹‘𝑞) ≤ 𝑘 ∧ 𝑘 ≤ (𝐹‘𝑟))))
114	111, 113	rspc2v 3622	. . . . . . . . . . . . . 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 7278	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐷–1-1-onto→(0[,)+∞) ∧ 𝑘 ∈ (0[,)+∞)) → (𝐹‘(^◡𝐹‘𝑘)) = 𝑘)
117	11, 89, 116	syl2an 595	. . . . . . . . . . . . . . 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 5160	. . . . . . . . . . . . . 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 5158	. . . . . . . . . . . . . 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 630	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 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 28664	. . . . . . . . . . . 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 467	. . . . . . . . . 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 3199	. . . . . . . . 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 4873	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑥 → ⟨𝑞, 𝑟⟩ = ⟨𝑥, 𝑟⟩)
127	126	breq2d 5160	. . . . . . . . . 10 ⊢ (𝑞 = 𝑥 → ((^◡𝐹‘𝑘) Btwn ⟨𝑞, 𝑟⟩ ↔ (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑟⟩))
128		opeq2 4874	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑦 → ⟨𝑥, 𝑟⟩ = ⟨𝑥, 𝑦⟩)
129	128	breq2d 5160	. . . . . . . . . 10 ⊢ (𝑟 = 𝑦 → ((^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑟⟩ ↔ (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑦⟩))
130	127, 129	cbvral2vw 3237	. . . . . . . . 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 5151	. . . . . . . . . 10 ⊢ (𝑏 = (^◡𝐹‘𝑘) → (𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑦⟩))
133	132	2ralbidv 3217	. . . . . . . . 9 ⊢ (𝑏 = (^◡𝐹‘𝑘) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (^◡𝐹‘𝑘) Btwn ⟨𝑥, 𝑦⟩))
134	133	rspcev 3612	. . . . . . . 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 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 ∈ (0[,)+∞))) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑏 Btwn ⟨𝑥, 𝑦⟩)
136	135	expr 456	. . . . . 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 412	. . . 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 3152	. 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 395 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3431 ⊆ wss 3948 ∅c0 4322 ⟨cop 4634 class class class wbr 5148 {copab 5210 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 “ cima 5679 Fun wfun 6537 ⟶wf 6539 –1-1→wf1 6540 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 +∞cpnf 11252 ≤ cle 11256 − cmin 11451 ℕcn 12219 [,)cico 13333 ...cfz 13491 𝔼cee 28581 Btwn cbtwn 28582

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-z 12566 df-uz 12830 df-ico 13337 df-icc 13338 df-fz 13492 df-ee 28584 df-btwn 28585

This theorem is referenced by: axcontlem11 28667

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