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Theorem pmltpclem2 25402

Description: Lemma for pmltpc 25403. (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 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → 𝑊 ∈ 𝐴)
3		pmltpc.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
4	3	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → 𝑈 ∈ 𝐴)
5		pmltpc.4	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
6	5	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → 𝑉 ∈ 𝐴)
7		simpr 484	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → 𝑊 < 𝑈)
8		pmltpc.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (ℝ ↑_pm ℝ))
9		reex 11220	. . . . . . . . . 10 ⊢ ℝ ∈ V
10	9, 9	elpm2 8888	. . . . . . . . 9 ⊢ (𝐹 ∈ (ℝ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
11	8, 10	sylib 218	. . . . . . . 8 ⊢ (𝜑 → (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
12	11	simprd 495	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
13		pmltpc.2	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
14	13, 3	sseldd 3959	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ dom 𝐹)
15	12, 14	sseldd 3959	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
16	13, 5	sseldd 3959	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ dom 𝐹)
17	12, 16	sseldd 3959	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℝ)
18		pmltpc.7	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ 𝑉)
19	11	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
20	19, 16	ffvelcdmd 7075	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑉) ∈ ℝ)
21		pmltpc.9	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐹‘𝑈) ≤ (𝐹‘𝑉))
22	19, 14	ffvelcdmd 7075	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑈) ∈ ℝ)
23	20, 22	ltnled 11382	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑉) < (𝐹‘𝑈) ↔ ¬ (𝐹‘𝑈) ≤ (𝐹‘𝑉)))
24	21, 23	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑉) < (𝐹‘𝑈))
25	20, 24	gtned 11370	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑈) ≠ (𝐹‘𝑉))
26		fveq2 6876	. . . . . . . . 9 ⊢ (𝑉 = 𝑈 → (𝐹‘𝑉) = (𝐹‘𝑈))
27	26	eqcomd 2741	. . . . . . . 8 ⊢ (𝑉 = 𝑈 → (𝐹‘𝑈) = (𝐹‘𝑉))
28	27	necon3i 2964	. . . . . . 7 ⊢ ((𝐹‘𝑈) ≠ (𝐹‘𝑉) → 𝑉 ≠ 𝑈)
29	25, 28	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
30	15, 17, 18, 29	leneltd 11389	. . . . 5 ⊢ (𝜑 → 𝑈 < 𝑉)
31	30	ad2antrr 726	. . . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → (𝐹‘𝑉) < (𝐹‘𝑈))
34	32, 33	jca 511	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑊 < 𝑈) → ((𝐹‘𝑊) < (𝐹‘𝑈) ∧ (𝐹‘𝑉) < (𝐹‘𝑈)))
35	34	orcd 873	. . . 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 25401	. . 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 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑈 ∈ 𝐴)
38	1	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑊 ∈ 𝐴)
39		pmltpc.6	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
40	39	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑋 ∈ 𝐴)
41	15	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑈 ∈ ℝ)
42	13, 1	sseldd 3959	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ dom 𝐹)
43	12, 42	sseldd 3959	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℝ)
44	43	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑊 ∈ ℝ)
45		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑈 ≤ 𝑊)
46	19, 42	ffvelcdmd 7075	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑊) ∈ ℝ)
47	46	ad2antrr 726	. . . . . . 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 11370	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → (𝐹‘𝑈) ≠ (𝐹‘𝑊))
50		fveq2 6876	. . . . . . . 8 ⊢ (𝑊 = 𝑈 → (𝐹‘𝑊) = (𝐹‘𝑈))
51	50	eqcomd 2741	. . . . . . 7 ⊢ (𝑊 = 𝑈 → (𝐹‘𝑈) = (𝐹‘𝑊))
52	51	necon3i 2964	. . . . . 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 11389	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑈 < 𝑊)
55	13, 39	sseldd 3959	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
56	12, 55	sseldd 3959	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ)
57		pmltpc.8	. . . . . 6 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
58		pmltpc.10	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐹‘𝑋) ≤ (𝐹‘𝑊))
59	19, 55	ffvelcdmd 7075	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑋) ∈ ℝ)
60	46, 59	ltnled 11382	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑊) < (𝐹‘𝑋) ↔ ¬ (𝐹‘𝑋) ≤ (𝐹‘𝑊)))
61	58, 60	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝑊) < (𝐹‘𝑋))
62	46, 61	gtned 11370	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑋) ≠ (𝐹‘𝑊))
63		fveq2 6876	. . . . . . . 8 ⊢ (𝑋 = 𝑊 → (𝐹‘𝑋) = (𝐹‘𝑊))
64	63	necon3i 2964	. . . . . . 7 ⊢ ((𝐹‘𝑋) ≠ (𝐹‘𝑊) → 𝑋 ≠ 𝑊)
65	62, 64	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 𝑊)
66	43, 56, 57, 65	leneltd 11389	. . . . 5 ⊢ (𝜑 → 𝑊 < 𝑋)
67	66	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → 𝑊 < 𝑋)
68	61	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → (𝐹‘𝑊) < (𝐹‘𝑋))
69	48, 68	jca 511	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) ∧ 𝑈 ≤ 𝑊) → ((𝐹‘𝑊) < (𝐹‘𝑈) ∧ (𝐹‘𝑊) < (𝐹‘𝑋)))
70	69	olcd 874	. . . 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 25401	. . 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 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) → 𝑊 ∈ ℝ)
73	15	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑊) < (𝐹‘𝑈)) → 𝑈 ∈ ℝ)
74	36, 71, 72, 73	ltlecasei 11343	. 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 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑈 ∈ 𝐴)
76	5	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑉 ∈ 𝐴)
77	39	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑋 ∈ 𝐴)
78	30	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑈 < 𝑉)
79		simpr 484	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → 𝑉 < 𝑋)
80	24	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → (𝐹‘𝑉) < (𝐹‘𝑈))
81	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑉) ∈ ℝ)
82	22	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑈) ∈ ℝ)
83	59	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑋) ∈ ℝ)
84	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑉) < (𝐹‘𝑈))
85	46	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑊) ∈ ℝ)
86		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑈) ≤ (𝐹‘𝑊))
87	61	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑊) < (𝐹‘𝑋))
88	82, 85, 83, 86, 87	lelttrd 11393	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑈) < (𝐹‘𝑋))
89	81, 82, 83, 84, 88	lttrd 11396	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → (𝐹‘𝑉) < (𝐹‘𝑋))
90	89	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → (𝐹‘𝑉) < (𝐹‘𝑋))
91	80, 90	jca 511	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑉 < 𝑋) → ((𝐹‘𝑉) < (𝐹‘𝑈) ∧ (𝐹‘𝑉) < (𝐹‘𝑋)))
92	91	olcd 874	. . . 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 25401	. . 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 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑊 ∈ 𝐴)
95	39	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑋 ∈ 𝐴)
96	5	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑉 ∈ 𝐴)
97	66	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑊 < 𝑋)
98	56	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑋 ∈ ℝ)
99	17	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑉 ∈ ℝ)
100		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑋 ≤ 𝑉)
101	20	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → (𝐹‘𝑉) ∈ ℝ)
102	89	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → (𝐹‘𝑉) < (𝐹‘𝑋))
103	101, 102	gtned 11370	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → (𝐹‘𝑋) ≠ (𝐹‘𝑉))
104		fveq2 6876	. . . . . . . 8 ⊢ (𝑉 = 𝑋 → (𝐹‘𝑉) = (𝐹‘𝑋))
105	104	eqcomd 2741	. . . . . . 7 ⊢ (𝑉 = 𝑋 → (𝐹‘𝑋) = (𝐹‘𝑉))
106	105	necon3i 2964	. . . . . 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 11389	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → 𝑋 < 𝑉)
109	61	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → (𝐹‘𝑊) < (𝐹‘𝑋))
110	109, 102	jca 511	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) ∧ 𝑋 ≤ 𝑉) → ((𝐹‘𝑊) < (𝐹‘𝑋) ∧ (𝐹‘𝑉) < (𝐹‘𝑋)))
111	110	orcd 873	. . . 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 25401	. . 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 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → 𝑉 ∈ ℝ)
114	56	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝑈) ≤ (𝐹‘𝑊)) → 𝑋 ∈ ℝ)
115	93, 112, 113, 114	ltlecasei 11343	. 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 11343	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 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∃wrex 3060 ⊆ wss 3926 class class class wbr 5119 dom cdm 5654 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ↑_pm cpm 8841 ℝcr 11128 < clt 11269 ≤ cle 11270

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-pre-lttri 11203 ax-pre-lttrn 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8719 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275

This theorem is referenced by: pmltpc 25403

W3C validator