MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vdwlem12 Structured version   Visualization version   GIF version

Theorem vdwlem12 15909
Description: Lemma for vdw 15911. 𝐾 = 2 base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
Hypotheses
Ref Expression
vdw.r (𝜑𝑅 ∈ Fin)
vdwlem12.f (𝜑𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
vdwlem12.2 (𝜑 → ¬ 2 MonoAP 𝐹)
Assertion
Ref Expression
vdwlem12 ¬ 𝜑

Proof of Theorem vdwlem12
Dummy variables 𝑎 𝑐 𝑑 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdw.r . . . . . . 7 (𝜑𝑅 ∈ Fin)
2 hashcl 13361 . . . . . . 7 (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ0)
31, 2syl 17 . . . . . 6 (𝜑 → (♯‘𝑅) ∈ ℕ0)
43nn0red 11614 . . . . 5 (𝜑 → (♯‘𝑅) ∈ ℝ)
54ltp1d 11236 . . . 4 (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
6 nn0p1nn 11594 . . . . . . 7 ((♯‘𝑅) ∈ ℕ0 → ((♯‘𝑅) + 1) ∈ ℕ)
73, 6syl 17 . . . . . 6 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
87nnnn0d 11613 . . . . 5 (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ0)
9 hashfz1 13350 . . . . 5 (((♯‘𝑅) + 1) ∈ ℕ0 → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
108, 9syl 17 . . . 4 (𝜑 → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
115, 10breqtrrd 4868 . . 3 (𝜑 → (♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))))
12 fzfi 12991 . . . 4 (1...((♯‘𝑅) + 1)) ∈ Fin
13 hashsdom 13384 . . . 4 ((𝑅 ∈ Fin ∧ (1...((♯‘𝑅) + 1)) ∈ Fin) → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
141, 12, 13sylancl 576 . . 3 (𝜑 → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
1511, 14mpbid 223 . 2 (𝜑𝑅 ≺ (1...((♯‘𝑅) + 1)))
16 vdwlem12.f . . . . 5 (𝜑𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
17 fveq2 6405 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
18 fveq2 6405 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐹𝑤) = (𝐹𝑦))
1917, 18eqeqan12d 2821 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
20 eqeq12 2818 . . . . . . . 8 ((𝑧 = 𝑥𝑤 = 𝑦) → (𝑧 = 𝑤𝑥 = 𝑦))
2119, 20imbi12d 335 . . . . . . 7 ((𝑧 = 𝑥𝑤 = 𝑦) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
22 fveq2 6405 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
23 fveq2 6405 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
2422, 23eqeqan12d 2821 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑦) = (𝐹𝑥)))
25 eqcom 2812 . . . . . . . . 9 ((𝐹𝑦) = (𝐹𝑥) ↔ (𝐹𝑥) = (𝐹𝑦))
2624, 25syl6bb 278 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑥) = (𝐹𝑦)))
27 eqeq12 2818 . . . . . . . . 9 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑦 = 𝑥))
28 eqcom 2812 . . . . . . . . 9 (𝑦 = 𝑥𝑥 = 𝑦)
2927, 28syl6bb 278 . . . . . . . 8 ((𝑧 = 𝑦𝑤 = 𝑥) → (𝑧 = 𝑤𝑥 = 𝑦))
3026, 29imbi12d 335 . . . . . . 7 ((𝑧 = 𝑦𝑤 = 𝑥) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
31 elfznn 12589 . . . . . . . . . 10 (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℕ)
3231nnred 11317 . . . . . . . . 9 (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℝ)
3332ssriv 3799 . . . . . . . 8 (1...((♯‘𝑅) + 1)) ⊆ ℝ
3433a1i 11 . . . . . . 7 (𝜑 → (1...((♯‘𝑅) + 1)) ⊆ ℝ)
35 biidd 253 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
36 simplr3 1272 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥𝑦)
37 vdwlem12.2 . . . . . . . . . . 11 (𝜑 → ¬ 2 MonoAP 𝐹)
3837ad2antrr 708 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 2 MonoAP 𝐹)
39 3simpa 1171 . . . . . . . . . . . 12 ((𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦) → (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1))))
40 simplrl 786 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (1...((♯‘𝑅) + 1)))
4140, 31syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℕ)
42 simprr 780 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
43 simplrr 787 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
44 elfznn 12589 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (1...((♯‘𝑅) + 1)) → 𝑦 ∈ ℕ)
4543, 44syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℕ)
46 nnsub 11341 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4741, 45, 46syl2anc 575 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 < 𝑦 ↔ (𝑦𝑥) ∈ ℕ))
4842, 47mpbid 223 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦𝑥) ∈ ℕ)
49 df-2 11360 . . . . . . . . . . . . . . . . . . 19 2 = (1 + 1)
5049fveq2i 6408 . . . . . . . . . . . . . . . . . 18 (AP‘2) = (AP‘(1 + 1))
5150oveqi 6884 . . . . . . . . . . . . . . . . 17 (𝑥(AP‘2)(𝑦𝑥)) = (𝑥(AP‘(1 + 1))(𝑦𝑥))
52 1nn0 11571 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ0
5352a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 1 ∈ ℕ0)
54 vdwapun 15891 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℕ0𝑥 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑥(AP‘(1 + 1))(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
5553, 41, 48, 54syl3anc 1483 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘(1 + 1))(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
5651, 55syl5eq 2851 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))))
57 simprl 778 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑥) = (𝐹𝑦))
5816ad2antrr 708 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
5958ffnd 6254 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((♯‘𝑅) + 1)))
60 fniniseg 6557 . . . . . . . . . . . . . . . . . . . 20 (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑥 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑥) = (𝐹𝑦))))
6159, 60syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑥) = (𝐹𝑦))))
6240, 57, 61mpbir2and 695 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (𝐹 “ {(𝐹𝑦)}))
6362snssd 4527 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (𝐹 “ {(𝐹𝑦)}))
6441nncnd 11318 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
6545nncnd 11318 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
6664, 65pncan3d 10677 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦𝑥)) = 𝑦)
6766oveq1d 6886 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) = (𝑦(AP‘1)(𝑦𝑥)))
68 vdwap1 15894 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
6945, 48, 68syl2anc 575 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦(AP‘1)(𝑦𝑥)) = {𝑦})
7067, 69eqtrd 2839 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) = {𝑦})
71 eqidd 2806 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝐹𝑦) = (𝐹𝑦))
72 fniniseg 6557 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑦 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑦) = (𝐹𝑦))))
7359, 72syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑦 ∈ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹𝑦) = (𝐹𝑦))))
7443, 71, 73mpbir2and 695 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (𝐹 “ {(𝐹𝑦)}))
7574snssd 4527 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (𝐹 “ {(𝐹𝑦)}))
7670, 75eqsstrd 3833 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
7763, 76unssd 3985 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ({𝑥} ∪ ((𝑥 + (𝑦𝑥))(AP‘1)(𝑦𝑥))) ⊆ (𝐹 “ {(𝐹𝑦)}))
7856, 77eqsstrd 3833 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)}))
79 oveq1 6878 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
8079sseq1d 3826 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
81 oveq2 6879 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑦𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦𝑥)))
8281sseq1d 3826 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑦𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) ↔ (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)})))
8380, 82rspc2ev 3516 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℕ ∧ (𝑦𝑥) ∈ ℕ ∧ (𝑥(AP‘2)(𝑦𝑥)) ⊆ (𝐹 “ {(𝐹𝑦)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}))
8441, 48, 78, 83syl3anc 1483 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}))
85 fvex 6418 . . . . . . . . . . . . . . 15 (𝐹𝑦) ∈ V
86 sneq 4377 . . . . . . . . . . . . . . . . . 18 (𝑐 = (𝐹𝑦) → {𝑐} = {(𝐹𝑦)})
8786imaeq2d 5673 . . . . . . . . . . . . . . . . 17 (𝑐 = (𝐹𝑦) → (𝐹 “ {𝑐}) = (𝐹 “ {(𝐹𝑦)}))
8887sseq2d 3827 . . . . . . . . . . . . . . . 16 (𝑐 = (𝐹𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
89882rexbidv 3244 . . . . . . . . . . . . . . 15 (𝑐 = (𝐹𝑦) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)})))
9085, 89spcev 3492 . . . . . . . . . . . . . 14 (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {(𝐹𝑦)}) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}))
9184, 90syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐}))
92 ovex 6903 . . . . . . . . . . . . . 14 (1...((♯‘𝑅) + 1)) ∈ V
93 2nn0 11572 . . . . . . . . . . . . . . 15 2 ∈ ℕ0
9493a1i 11 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℕ0)
9592, 94, 58vdwmc 15895 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → (2 MonoAP 𝐹 ↔ ∃𝑐𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (𝐹 “ {𝑐})))
9691, 95mpbird 248 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9739, 96sylanl2 663 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
9897expr 446 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 < 𝑦 → 2 MonoAP 𝐹))
9938, 98mtod 189 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → ¬ 𝑥 < 𝑦)
100 simplr1 1268 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 ∈ (1...((♯‘𝑅) + 1)))
101100, 32syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 ∈ ℝ)
102 simplr2 1270 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
10333, 102sseldi 3793 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑦 ∈ ℝ)
104101, 103eqleltd 10463 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥 = 𝑦 ↔ (𝑥𝑦 ∧ ¬ 𝑥 < 𝑦)))
10536, 99, 104mpbir2and 695 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 = 𝑦)
106105ex 399 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥𝑦)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
10721, 30, 34, 35, 106wlogle 10843 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
108107ralrimivva 3158 . . . . 5 (𝜑 → ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
109 dff13 6733 . . . . 5 (𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅 ↔ (𝐹:(1...((♯‘𝑅) + 1))⟶𝑅 ∧ ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
11016, 108, 109sylanbrc 574 . . . 4 (𝜑𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅)
111 f1domg 8209 . . . 4 (𝑅 ∈ Fin → (𝐹:(1...((♯‘𝑅) + 1))–1-1𝑅 → (1...((♯‘𝑅) + 1)) ≼ 𝑅))
1121, 110, 111sylc 65 . . 3 (𝜑 → (1...((♯‘𝑅) + 1)) ≼ 𝑅)
113 domnsym 8322 . . 3 ((1...((♯‘𝑅) + 1)) ≼ 𝑅 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
114112, 113syl 17 . 2 (𝜑 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
11515, 114pm2.65i 185 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2158  wral 3095  wrex 3096  cun 3764  wss 3766  {csn 4367   class class class wbr 4840  ccnv 5307  cima 5311   Fn wfn 6093  wf 6094  1-1wf1 6095  cfv 6098  (class class class)co 6871  cdom 8187  csdm 8188  Fincfn 8189  cr 10217  1c1 10219   + caddc 10221   < clt 10356  cle 10357  cmin 10548  cn 11302  2c2 11352  0cn0 11555  ...cfz 12545  chash 13333  APcvdwa 15882   MonoAP cvdwm 15883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176  ax-cnex 10274  ax-resscn 10275  ax-1cn 10276  ax-icn 10277  ax-addcl 10278  ax-addrcl 10279  ax-mulcl 10280  ax-mulrcl 10281  ax-mulcom 10282  ax-addass 10283  ax-mulass 10284  ax-distr 10285  ax-i2m1 10286  ax-1ne0 10287  ax-1rid 10288  ax-rnegex 10289  ax-rrecex 10290  ax-cnre 10291  ax-pre-lttri 10292  ax-pre-lttrn 10293  ax-pre-ltadd 10294  ax-pre-mulgt0 10295
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-reu 3102  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-oadd 7797  df-er 7976  df-en 8190  df-dom 8191  df-sdom 8192  df-fin 8193  df-card 9045  df-pnf 10358  df-mnf 10359  df-xr 10360  df-ltxr 10361  df-le 10362  df-sub 10550  df-neg 10551  df-nn 11303  df-2 11360  df-n0 11556  df-xnn0 11626  df-z 11640  df-uz 11901  df-fz 12546  df-hash 13334  df-vdwap 15885  df-vdwmc 15886
This theorem is referenced by:  vdwlem13  15910
  Copyright terms: Public domain W3C validator