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Theorem pmltpclem2 24053

Description: Lemma for pmltpc 24054. (Contributed by Mario Carneiro, 1-Jul-2014.)

**Hypotheses**
Ref	Expression
pmltpc.1	⊢ (𝜑 → 𝐹 ∈ (ℝ ↑_pm ℝ))
pmltpc.2	⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
pmltpc.3	⊢ (𝜑 → 𝑈 ∈ 𝐴)
pmltpc.4	⊢ (𝜑 → 𝑉 ∈ 𝐴)
pmltpc.5	⊢ (𝜑 → 𝑊 ∈ 𝐴)
pmltpc.6	⊢ (𝜑 → 𝑋 ∈ 𝐴)
pmltpc.7	⊢ (𝜑 → 𝑈 ≤ 𝑉)
pmltpc.8	⊢ (𝜑 → 𝑊 ≤ 𝑋)
pmltpc.9	⊢ (𝜑 → ¬ (𝐹‘𝑈) ≤ (𝐹‘𝑉))
pmltpc.10	⊢ (𝜑 → ¬ (𝐹‘𝑋) ≤ (𝐹‘𝑊))

**Assertion**
Ref	Expression
pmltpclem2	⊢ (𝜑 → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))

Distinct variable groups: 𝑎,𝑏,𝑐,𝐴 𝐹,𝑎,𝑏,𝑐 𝑉,𝑏,𝑐 𝑈,𝑎,𝑏,𝑐 𝑊,𝑎,𝑏,𝑐 𝑋,𝑏,𝑐 Allowed substitution hints: 𝜑(𝑎,𝑏,𝑐) 𝑉(𝑎) 𝑋(𝑎)

**Proof of Theorem pmltpclem2**
Step	Hyp	Ref	Expression
1		pmltpc.5	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐴)
2	1	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → 𝑊 ∈ 𝐴)
3		pmltpc.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
4	3	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → 𝑈 ∈ 𝐴)
5		pmltpc.4	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
6	5	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → 𝑉 ∈ 𝐴)
7		simpr 488	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → 𝑊 < 𝑈)
8		pmltpc.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑_pm ℝ))
9		reex 10617	. . . . . . . . . 10 ⊢ ℝ ∈ V
10	9, 9	elpm2 8421	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℝ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
11	8, 10	sylib 221	. . . . . . . 8 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
12	11	simprd 499	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
13		pmltpc.2	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
14	13, 3	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ dom 𝐹)
15	12, 14	sseldd 3916	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
16	13, 5	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ dom 𝐹)
17	12, 16	sseldd 3916	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ)
18		pmltpc.7	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ 𝑉)
19	11	simpld 498	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
20	19, 16	ffvelrnd 6829	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑉) ∈ ℝ)
21		pmltpc.9	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐹‘𝑈) ≤ (𝐹‘𝑉))
22	19, 14	ffvelrnd 6829	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑈) ∈ ℝ)
23	20, 22	ltnled 10776	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑉) < (𝐹‘𝑈) ↔ ¬ (𝐹‘𝑈) ≤ (𝐹‘𝑉)))
24	21, 23	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑉) < (𝐹‘𝑈))
25	20, 24	gtned 10764	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑈) ≠ (𝐹‘𝑉))
26		fveq2 6645	. . . . . . . . 9 ⊢ (𝑉 = 𝑈 → (𝐹‘𝑉) = (𝐹‘𝑈))
27	26	eqcomd 2804	. . . . . . . 8 ⊢ (𝑉 = 𝑈 → (𝐹‘𝑈) = (𝐹‘𝑉))
28	27	necon3i 3019	. . . . . . 7 ⊢ ((𝐹‘𝑈) ≠ (𝐹‘𝑉) → 𝑉 ≠ 𝑈)
29	25, 28	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
30	15, 17, 18, 29	leneltd 10783	. . . . 5 ⊢ (𝜑 → 𝑈 < 𝑉)
31	30	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → 𝑈 < 𝑉)
32		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → (𝐹‘𝑊) < (𝐹‘𝑈))
33	24	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → (𝐹‘𝑉) < (𝐹‘𝑈))
34	32, 33	jca 515	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → ((𝐹‘𝑊) < (𝐹‘𝑈) ∧ (𝐹‘𝑉) < (𝐹‘𝑈)))
35	34	orcd 870	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → (((𝐹‘𝑊) < (𝐹‘𝑈) ∧ (𝐹‘𝑉) < (𝐹‘𝑈)) ∨ ((𝐹‘𝑈) < (𝐹‘𝑊) ∧ (𝐹‘𝑈) < (𝐹‘𝑉))))
36	2, 4, 6, 7, 31, 35	pmltpclem1 24052	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
37	3	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑈 ∈ 𝐴)
38	1	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑊 ∈ 𝐴)
39		pmltpc.6	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
40	39	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑋 ∈ 𝐴)
41	15	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑈 ∈ ℝ)
42	13, 1	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ dom 𝐹)
43	12, 42	sseldd 3916	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℝ)
44	43	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑊 ∈ ℝ)
45		simpr 488	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑈 ≤ 𝑊)
46	19, 42	ffvelrnd 6829	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑊) ∈ ℝ)
47	46	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → (𝐹‘𝑊) ∈ ℝ)
48		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → (𝐹‘𝑊) < (𝐹‘𝑈))
49	47, 48	gtned 10764	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → (𝐹‘𝑈) ≠ (𝐹‘𝑊))
50		fveq2 6645	. . . . . . . 8 ⊢ (𝑊 = 𝑈 → (𝐹‘𝑊) = (𝐹‘𝑈))
51	50	eqcomd 2804	. . . . . . 7 ⊢ (𝑊 = 𝑈 → (𝐹‘𝑈) = (𝐹‘𝑊))
52	51	necon3i 3019	. . . . . 6 ⊢ ((𝐹‘𝑈) ≠ (𝐹‘𝑊) → 𝑊 ≠ 𝑈)
53	49, 52	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑊 ≠ 𝑈)
54	41, 44, 45, 53	leneltd 10783	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑈 < 𝑊)
55	13, 39	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
56	12, 55	sseldd 3916	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ)
57		pmltpc.8	. . . . . 6 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
58		pmltpc.10	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐹‘𝑋) ≤ (𝐹‘𝑊))
59	19, 55	ffvelrnd 6829	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ)
60	46, 59	ltnled 10776	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑊) < (𝐹‘𝑋) ↔ ¬ (𝐹‘𝑋) ≤ (𝐹‘𝑊)))
61	58, 60	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑊) < (𝐹‘𝑋))
62	46, 61	gtned 10764	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑋) ≠ (𝐹‘𝑊))
63		fveq2 6645	. . . . . . . 8 ⊢ (𝑋 = 𝑊 → (𝐹‘𝑋) = (𝐹‘𝑊))
64	63	necon3i 3019	. . . . . . 7 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑊) → 𝑋 ≠ 𝑊)
65	62, 64	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑊)
66	43, 56, 57, 65	leneltd 10783	. . . . 5 ⊢ (𝜑 → 𝑊 < 𝑋)
67	66	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑊 < 𝑋)
68	61	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → (𝐹‘𝑊) < (𝐹‘𝑋))
69	48, 68	jca 515	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → ((𝐹‘𝑊) < (𝐹‘𝑈) ∧ (𝐹‘𝑊) < (𝐹‘𝑋)))
70	69	olcd 871	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → (((𝐹‘𝑈) < (𝐹‘𝑊) ∧ (𝐹‘𝑋) < (𝐹‘𝑊)) ∨ ((𝐹‘𝑊) < (𝐹‘𝑈) ∧ (𝐹‘𝑊) < (𝐹‘𝑋))))
71	37, 38, 40, 54, 67, 70	pmltpclem1 24052	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
72	43	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) → 𝑊 ∈ ℝ)
73	15	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) → 𝑈 ∈ ℝ)
74	36, 71, 72, 73	ltlecasei 10737	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
75	3	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑈 ∈ 𝐴)
76	5	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑉 ∈ 𝐴)
77	39	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑋 ∈ 𝐴)
78	30	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑈 < 𝑉)
79		simpr 488	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑉 < 𝑋)
80	24	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → (𝐹‘𝑉) < (𝐹‘𝑈))
81	20	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑉) ∈ ℝ)
82	22	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑈) ∈ ℝ)
83	59	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑋) ∈ ℝ)
84	24	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑉) < (𝐹‘𝑈))
85	46	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑊) ∈ ℝ)
86		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑈) ≤ (𝐹‘𝑊))
87	61	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑊) < (𝐹‘𝑋))
88	82, 85, 83, 86, 87	lelttrd 10787	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑈) < (𝐹‘𝑋))
89	81, 82, 83, 84, 88	lttrd 10790	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑉) < (𝐹‘𝑋))
90	89	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → (𝐹‘𝑉) < (𝐹‘𝑋))
91	80, 90	jca 515	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → ((𝐹‘𝑉) < (𝐹‘𝑈) ∧ (𝐹‘𝑉) < (𝐹‘𝑋)))
92	91	olcd 871	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → (((𝐹‘𝑈) < (𝐹‘𝑉) ∧ (𝐹‘𝑋) < (𝐹‘𝑉)) ∨ ((𝐹‘𝑉) < (𝐹‘𝑈) ∧ (𝐹‘𝑉) < (𝐹‘𝑋))))
93	75, 76, 77, 78, 79, 92	pmltpclem1 24052	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
94	1	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑊 ∈ 𝐴)
95	39	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑋 ∈ 𝐴)
96	5	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑉 ∈ 𝐴)
97	66	ad2antrr 725	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑊 < 𝑋)
98	56	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑋 ∈ ℝ)
99	17	ad2antrr 725	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑉 ∈ ℝ)
100		simpr 488	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑋 ≤ 𝑉)
101	20	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → (𝐹‘𝑉) ∈ ℝ)
102	89	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → (𝐹‘𝑉) < (𝐹‘𝑋))
103	101, 102	gtned 10764	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → (𝐹‘𝑋) ≠ (𝐹‘𝑉))
104		fveq2 6645	. . . . . . . 8 ⊢ (𝑉 = 𝑋 → (𝐹‘𝑉) = (𝐹‘𝑋))
105	104	eqcomd 2804	. . . . . . 7 ⊢ (𝑉 = 𝑋 → (𝐹‘𝑋) = (𝐹‘𝑉))
106	105	necon3i 3019	. . . . . 6 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑉) → 𝑉 ≠ 𝑋)
107	103, 106	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑉 ≠ 𝑋)
108	98, 99, 100, 107	leneltd 10783	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑋 < 𝑉)
109	61	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → (𝐹‘𝑊) < (𝐹‘𝑋))
110	109, 102	jca 515	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → ((𝐹‘𝑊) < (𝐹‘𝑋) ∧ (𝐹‘𝑉) < (𝐹‘𝑋)))
111	110	orcd 870	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → (((𝐹‘𝑊) < (𝐹‘𝑋) ∧ (𝐹‘𝑉) < (𝐹‘𝑋)) ∨ ((𝐹‘𝑋) < (𝐹‘𝑊) ∧ (𝐹‘𝑋) < (𝐹‘𝑉))))
112	94, 95, 96, 97, 108, 111	pmltpclem1 24052	. . 3 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
113	17	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → 𝑉 ∈ ℝ)
114	56	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → 𝑋 ∈ ℝ)
115	93, 112, 113, 114	ltlecasei 10737	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
116	74, 115, 46, 22	ltlecasei 10737	1 ⊢ (𝜑 → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ 𝐴 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ⊆ wss 3881 class class class wbr 5030 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑_pm cpm 8390 ℝcr 10525 < clt 10664 ≤ cle 10665

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670

This theorem is referenced by: pmltpc 24054

W3C validator