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Theorem fphpdo 38899
Description: Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)
Hypotheses
Ref Expression
fphpdo.1 (𝜑𝐴 ⊆ ℝ)
fphpdo.2 (𝜑𝐵 ∈ V)
fphpdo.3 (𝜑𝐵𝐴)
fphpdo.4 ((𝜑𝑧𝐴) → 𝐶𝐵)
fphpdo.5 (𝑧 = 𝑥𝐶 = 𝐷)
fphpdo.6 (𝑧 = 𝑦𝐶 = 𝐸)
Assertion
Ref Expression
fphpdo (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸))
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑧,𝐵   𝑥,𝐶,𝑦   𝑦,𝐷,𝑧   𝑥,𝐸,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑧)   𝐷(𝑥)   𝐸(𝑦)

Proof of Theorem fphpdo
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fphpdo.3 . . 3 (𝜑𝐵𝐴)
2 fphpdo.4 . . . . 5 ((𝜑𝑧𝐴) → 𝐶𝐵)
32fmpttd 6742 . . . 4 (𝜑 → (𝑧𝐴𝐶):𝐴𝐵)
43ffvelrnda 6716 . . 3 ((𝜑𝑏𝐴) → ((𝑧𝐴𝐶)‘𝑏) ∈ 𝐵)
5 fveq2 6538 . . 3 (𝑏 = 𝑐 → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
61, 4, 5fphpd 38898 . 2 (𝜑 → ∃𝑏𝐴𝑐𝐴 (𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)))
7 fphpdo.1 . . . . . . . . . 10 (𝜑𝐴 ⊆ ℝ)
87sselda 3889 . . . . . . . . 9 ((𝜑𝑏𝐴) → 𝑏 ∈ ℝ)
98adantrr 713 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑏 ∈ ℝ)
109adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → 𝑏 ∈ ℝ)
117sselda 3889 . . . . . . . . 9 ((𝜑𝑐𝐴) → 𝑐 ∈ ℝ)
1211adantrl 712 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑐 ∈ ℝ)
1312adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → 𝑐 ∈ ℝ)
1410, 13lttri2d 10626 . . . . . 6 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏𝑐 ↔ (𝑏 < 𝑐𝑐 < 𝑏)))
15 simprl 767 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑏𝐴)
1615ad2antrr 722 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑏𝐴)
17 simprr 769 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑐𝐴)
1817ad2antrr 722 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑐𝐴)
19 simpr 485 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑏 < 𝑐)
20 simplr 765 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
21 breq1 4965 . . . . . . . . . . . 12 (𝑥 = 𝑏 → (𝑥 < 𝑦𝑏 < 𝑦))
22 fveqeq2 6547 . . . . . . . . . . . 12 (𝑥 = 𝑏 → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦)))
2321, 22anbi12d 630 . . . . . . . . . . 11 (𝑥 = 𝑏 → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑏 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦))))
24 breq2 4966 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (𝑏 < 𝑦𝑏 < 𝑐))
25 fveq2 6538 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → ((𝑧𝐴𝐶)‘𝑦) = ((𝑧𝐴𝐶)‘𝑐))
2625eqeq2d 2805 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)))
2724, 26anbi12d 630 . . . . . . . . . . 11 (𝑦 = 𝑐 → ((𝑏 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑏 < 𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))))
2823, 27rspc2ev 3574 . . . . . . . . . 10 ((𝑏𝐴𝑐𝐴 ∧ (𝑏 < 𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
2916, 18, 19, 20, 28syl112anc 1367 . . . . . . . . 9 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
3029ex 413 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏 < 𝑐 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
3117ad2antrr 722 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑐𝐴)
3215ad2antrr 722 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑏𝐴)
33 simpr 485 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑐 < 𝑏)
34 simplr 765 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
3534eqcomd 2801 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))
36 breq1 4965 . . . . . . . . . . . 12 (𝑥 = 𝑐 → (𝑥 < 𝑦𝑐 < 𝑦))
37 fveqeq2 6547 . . . . . . . . . . . 12 (𝑥 = 𝑐 → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦)))
3836, 37anbi12d 630 . . . . . . . . . . 11 (𝑥 = 𝑐 → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑐 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦))))
39 breq2 4966 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (𝑐 < 𝑦𝑐 < 𝑏))
40 fveq2 6538 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → ((𝑧𝐴𝐶)‘𝑦) = ((𝑧𝐴𝐶)‘𝑏))
4140eqeq2d 2805 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏)))
4239, 41anbi12d 630 . . . . . . . . . . 11 (𝑦 = 𝑏 → ((𝑐 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑐 < 𝑏 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))))
4338, 42rspc2ev 3574 . . . . . . . . . 10 ((𝑐𝐴𝑏𝐴 ∧ (𝑐 < 𝑏 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
4431, 32, 33, 35, 43syl112anc 1367 . . . . . . . . 9 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
4544ex 413 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑐 < 𝑏 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
4630, 45jaod 854 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ((𝑏 < 𝑐𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
47 eqid 2795 . . . . . . . . . . . . . 14 (𝑧𝐴𝐶) = (𝑧𝐴𝐶)
48 fphpdo.5 . . . . . . . . . . . . . 14 (𝑧 = 𝑥𝐶 = 𝐷)
49 simplr 765 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑥𝐴)
50 eleq1w 2865 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
5150anbi2d 628 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → ((𝜑𝑧𝐴) ↔ (𝜑𝑥𝐴)))
5248eleq1d 2867 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝐶𝐵𝐷𝐵))
5351, 52imbi12d 346 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (((𝜑𝑧𝐴) → 𝐶𝐵) ↔ ((𝜑𝑥𝐴) → 𝐷𝐵)))
5453, 2chvarv 2370 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝐷𝐵)
5554adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝐷𝐵)
5647, 48, 49, 55fvmptd3 6657 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑧𝐴𝐶)‘𝑥) = 𝐷)
57 fphpdo.6 . . . . . . . . . . . . . 14 (𝑧 = 𝑦𝐶 = 𝐸)
58 simpr 485 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑦𝐴)
59 eleq1w 2865 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
6059anbi2d 628 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → ((𝜑𝑧𝐴) ↔ (𝜑𝑦𝐴)))
6157eleq1d 2867 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝐶𝐵𝐸𝐵))
6260, 61imbi12d 346 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → (((𝜑𝑧𝐴) → 𝐶𝐵) ↔ ((𝜑𝑦𝐴) → 𝐸𝐵)))
6362, 2chvarv 2370 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐴) → 𝐸𝐵)
6463adantlr 711 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝐸𝐵)
6547, 57, 58, 64fvmptd3 6657 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑧𝐴𝐶)‘𝑦) = 𝐸)
6656, 65eqeq12d 2810 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ 𝐷 = 𝐸))
6766biimpd 230 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) → 𝐷 = 𝐸))
6867anim2d 611 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → (𝑥 < 𝑦𝐷 = 𝐸)))
6968reximdva 3237 . . . . . . . . 9 ((𝜑𝑥𝐴) → (∃𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7069reximdva 3237 . . . . . . . 8 (𝜑 → (∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7170ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7246, 71syld 47 . . . . . 6 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ((𝑏 < 𝑐𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7314, 72sylbid 241 . . . . 5 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏𝑐 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7473expimpd 454 . . . 4 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → ((((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐) ∧ 𝑏𝑐) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7574ancomsd 466 . . 3 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → ((𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7675rexlimdvva 3257 . 2 (𝜑 → (∃𝑏𝐴𝑐𝐴 (𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
776, 76mpd 15 1 (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 842   = wceq 1522  wcel 2081  wne 2984  wrex 3106  Vcvv 3437  wss 3859   class class class wbr 4962  cmpt 5041  cfv 6225  csdm 8356  cr 10382   < clt 10521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-resscn 10440  ax-pre-lttri 10457  ax-pre-lttrn 10458
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-po 5362  df-so 5363  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-ltxr 10526
This theorem is referenced by:  irrapxlem1  38904
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