Theorem pmltpclem1 24051

Description: Lemma for pmltpc 24053. (Contributed by Mario Carneiro, 1-Jul-2014.)

Hypotheses

Ref	Expression
pmltpclem1.1	⊢ (𝜑 → 𝐴 ∈ 𝑆)
pmltpclem1.2	⊢ (𝜑 → 𝐵 ∈ 𝑆)
pmltpclem1.3	⊢ (𝜑 → 𝐶 ∈ 𝑆)
pmltpclem1.4	⊢ (𝜑 → 𝐴 < 𝐵)
pmltpclem1.5	⊢ (𝜑 → 𝐵 < 𝐶)
pmltpclem1.6	⊢ (𝜑 → (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝐶) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝐶))))

Assertion

Ref	Expression
pmltpclem1	⊢ (𝜑 → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 (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   𝐵,𝑏,𝑐   𝐶,𝑐   𝐹,𝑎,𝑏,𝑐   𝑆,𝑎,𝑏,𝑐

Allowed substitution hints:   𝜑(𝑎,𝑏,𝑐)   𝐵(𝑎)   𝐶(𝑎,𝑏)

Proof of Theorem pmltpclem1

Step	Hyp	Ref	Expression
1		pmltpclem1.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
2		pmltpclem1.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
3		pmltpclem1.3	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
4		pmltpclem1.4	. 2 ⊢ (𝜑 → 𝐴 < 𝐵)
5		pmltpclem1.5	. 2 ⊢ (𝜑 → 𝐵 < 𝐶)
6		pmltpclem1.6	. 2 ⊢ (𝜑 → (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝐶) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝐶))))
7		breq1 5071	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎 < 𝑏 ↔ 𝐴 < 𝑏))
8		fveq2 6672	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
9	8	breq1d 5078	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) < (𝐹‘𝑏) ↔ (𝐹‘𝐴) < (𝐹‘𝑏)))
10	9	anbi1d 631	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ↔ ((𝐹‘𝐴) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏))))
11	8	breq2d 5080	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑏) < (𝐹‘𝑎) ↔ (𝐹‘𝑏) < (𝐹‘𝐴)))
12	11	anbi1d 631	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)) ↔ ((𝐹‘𝑏) < (𝐹‘𝐴) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))
13	10, 12	orbi12d 915	. . . 4 ⊢ (𝑎 = 𝐴 → ((((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))) ↔ (((𝐹‘𝐴) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝐴) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
14	7, 13	3anbi13d 1434	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))) ↔ (𝐴 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝐴) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝐴) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))))))
15		breq2 5072	. . . 4 ⊢ (𝑏 = 𝐵 → (𝐴 < 𝑏 ↔ 𝐴 < 𝐵))
16		breq1 5071	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑏 < 𝑐 ↔ 𝐵 < 𝑐))
17		fveq2 6672	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
18	17	breq2d 5080	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝐴) < (𝐹‘𝑏) ↔ (𝐹‘𝐴) < (𝐹‘𝐵)))
19	17	breq2d 5080	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝑐) < (𝐹‘𝑏) ↔ (𝐹‘𝑐) < (𝐹‘𝐵)))
20	18, 19	anbi12d 632	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐹‘𝐴) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ↔ ((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝑐) < (𝐹‘𝐵))))
21	17	breq1d 5078	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝑏) < (𝐹‘𝐴) ↔ (𝐹‘𝐵) < (𝐹‘𝐴)))
22	17	breq1d 5078	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝑏) < (𝐹‘𝑐) ↔ (𝐹‘𝐵) < (𝐹‘𝑐)))
23	21, 22	anbi12d 632	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐹‘𝑏) < (𝐹‘𝐴) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)) ↔ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝑐))))
24	20, 23	orbi12d 915	. . . 4 ⊢ (𝑏 = 𝐵 → ((((𝐹‘𝐴) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝐴) ∧ (𝐹‘𝑏) < (𝐹‘𝑐))) ↔ (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝑐) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝑐)))))
25	15, 16, 24	3anbi123d 1432	. . 3 ⊢ (𝑏 = 𝐵 → ((𝐴 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝐴) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝐴) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))) ↔ (𝐴 < 𝐵 ∧ 𝐵 < 𝑐 ∧ (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝑐) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝑐))))))
26		breq2 5072	. . . 4 ⊢ (𝑐 = 𝐶 → (𝐵 < 𝑐 ↔ 𝐵 < 𝐶))
27		fveq2 6672	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐹‘𝑐) = (𝐹‘𝐶))
28	27	breq1d 5078	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝐹‘𝑐) < (𝐹‘𝐵) ↔ (𝐹‘𝐶) < (𝐹‘𝐵)))
29	28	anbi2d 630	. . . . 5 ⊢ (𝑐 = 𝐶 → (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝑐) < (𝐹‘𝐵)) ↔ ((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝐶) < (𝐹‘𝐵))))
30	27	breq2d 5080	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝐹‘𝐵) < (𝐹‘𝑐) ↔ (𝐹‘𝐵) < (𝐹‘𝐶)))
31	30	anbi2d 630	. . . . 5 ⊢ (𝑐 = 𝐶 → (((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝑐)) ↔ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝐶))))
32	29, 31	orbi12d 915	. . . 4 ⊢ (𝑐 = 𝐶 → ((((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝑐) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝑐))) ↔ (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝐶) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝐶)))))
33	26, 32	3anbi23d 1435	. . 3 ⊢ (𝑐 = 𝐶 → ((𝐴 < 𝐵 ∧ 𝐵 < 𝑐 ∧ (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝑐) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝑐)))) ↔ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ∧ (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝐶) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝐶))))))
34	14, 25, 33	rspc3ev 3639	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ∧ (((𝐹‘𝐴) < (𝐹‘𝐵) ∧ (𝐹‘𝐶) < (𝐹‘𝐵)) ∨ ((𝐹‘𝐵) < (𝐹‘𝐴) ∧ (𝐹‘𝐵) < (𝐹‘𝐶))))) → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))
35	1, 2, 3, 4, 5, 6, 34	syl33anc 1381	1 ⊢ (𝜑 → ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 (𝑎 < 𝑏 ∧ 𝑏 < 𝑐 ∧ (((𝐹‘𝑎) < (𝐹‘𝑏) ∧ (𝐹‘𝑐) < (𝐹‘𝑏)) ∨ ((𝐹‘𝑏) < (𝐹‘𝑎) ∧ (𝐹‘𝑏) < (𝐹‘𝑐)))))

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃wrex 3141   class class class wbr 5068  ‘cfv 6357   < clt 10677

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365

This theorem is referenced by:  pmltpclem2  24052