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Theorem vdwlem12 17047

Description: Lemma for vdw 17049. 𝐾 = 2 base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)

**Hypotheses**
Ref	Expression
vdw.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem12.f	⊢ (𝜑 → 𝐹:(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.

Step	Hyp	Ref	Expression
1		vdw.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Fin)
2		hashcl 14388	. . . . . . 7 ⊢ (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ₀)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘𝑅) ∈ ℕ₀)
4	3	nn0red 12561	. . . . 5 ⊢ (𝜑 → (♯‘𝑅) ∈ ℝ)
5	4	ltp1d 12140	. . . 4 ⊢ (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
6		nn0p1nn 12538	. . . . . . 7 ⊢ ((♯‘𝑅) ∈ ℕ₀ → ((♯‘𝑅) + 1) ∈ ℕ)
7	3, 6	syl 18	. . . . . 6 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
8	7	nnnn0d 12560	. . . . 5 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ₀)
9		hashfz1 14378	. . . . 5 ⊢ (((♯‘𝑅) + 1) ∈ ℕ₀ → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → (♯‘(1...((♯‘𝑅) + 1))) = ((♯‘𝑅) + 1))
11	5, 10	breqtrrd 5139	. . 3 ⊢ (𝜑 → (♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))))
12		fzfi 14004	. . . 4 ⊢ (1...((♯‘𝑅) + 1)) ∈ Fin
13		hashsdom 14413	. . . 4 ⊢ ((𝑅 ∈ Fin ∧ (1...((♯‘𝑅) + 1)) ∈ Fin) → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
14	1, 12, 13	sylancl 597	. . 3 ⊢ (𝜑 → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
15	11, 14	mpbid 235	. 2 ⊢ (𝜑 → 𝑅 ≺ (1...((♯‘𝑅) + 1)))
16		vdwlem12.f	. . . . 5 ⊢ (𝜑 → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
17		fveq2 6881	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
18		fveq2 6881	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦))
19	17, 18	eqeqan12d 2777	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
20		eqeq12 2780	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝑧 = 𝑤 ↔ 𝑥 = 𝑦))
21	19, 20	imbi12d 347	. . . . . . 7 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
22		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
23		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
24	22, 23	eqeqan12d 2777	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
25		eqcom 2770	. . . . . . . . 9 ⊢ ((𝐹‘𝑦) = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = (𝐹‘𝑦))
26	24, 25	bitrdi 290	. . . . . . . 8 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
27		eqeq12 2780	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑧 = 𝑤 ↔ 𝑦 = 𝑥))
28		eqcom 2770	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
29	27, 28	bitrdi 290	. . . . . . . 8 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑧 = 𝑤 ↔ 𝑥 = 𝑦))
30	26, 29	imbi12d 347	. . . . . . 7 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
31		elfznn 13577	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℕ)
32	31	nnred 12243	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℝ)
33	32	ssriv 3941	. . . . . . . 8 ⊢ (1...((♯‘𝑅) + 1)) ⊆ ℝ
34	33	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1...((♯‘𝑅) + 1)) ⊆ ℝ)
35		biidd 265	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
36		simplr3 1236	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 ≤ 𝑦)
37		vdwlem12.2	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 2 MonoAP 𝐹)
38	37	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ¬ 2 MonoAP 𝐹)
39		3simpa 1166	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦) → (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1))))
40		simplrl 788	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (1...((♯‘𝑅) + 1)))
41	40, 31	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℕ)
42		simprr 784	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
43		simplrr 789	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
44		elfznn 13577	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1...((♯‘𝑅) + 1)) → 𝑦 ∈ ℕ)
45	43, 44	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℕ)
46		nnsub 12275	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℕ))
47	41, 45, 46	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℕ))
48	42, 47	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑦 − 𝑥) ∈ ℕ)
49		df-2 12298	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 = (1 + 1)
50	49	fveq2i 6884	. . . . . . . . . . . . . . . . . 18 ⊢ (AP‘2) = (AP‘(1 + 1))
51	50	oveqi 7423	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(AP‘2)(𝑦 − 𝑥)) = (𝑥(AP‘(1 + 1))(𝑦 − 𝑥))
52		1nn0 12515	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
53		vdwapun 17029	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℕ₀ ∧ 𝑥 ∈ ℕ ∧ (𝑦 − 𝑥) ∈ ℕ) → (𝑥(AP‘(1 + 1))(𝑦 − 𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦 − 𝑥))(AP‘1)(𝑦 − 𝑥))))
54	52, 41, 48, 53	mp3an2i 1495	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘(1 + 1))(𝑦 − 𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦 − 𝑥))(AP‘1)(𝑦 − 𝑥))))
55	51, 54	eqtrid 2810	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦 − 𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦 − 𝑥))(AP‘1)(𝑦 − 𝑥))))
56		simprl 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) = (𝐹‘𝑦))
57	16	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
58	57	ffnd 6706	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((♯‘𝑅) + 1)))
59		fniniseg 7055	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑥 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))))
60	58, 59	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))))
61	40, 56, 60	mpbir2and 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}))
62	61	snssd 4752	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
63	41	nncnd 12244	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
64	45	nncnd 12244	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
65	63, 64	pncan3d 11567	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦 − 𝑥)) = 𝑦)
66	65	oveq1d 7425	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦 − 𝑥))(AP‘1)(𝑦 − 𝑥)) = (𝑦(AP‘1)(𝑦 − 𝑥)))
67		vdwap1 17032	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 − 𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦 − 𝑥)) = {𝑦})
68	45, 48, 67	syl2anc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑦(AP‘1)(𝑦 − 𝑥)) = {𝑦})
69	66, 68	eqtrd 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦 − 𝑥))(AP‘1)(𝑦 − 𝑥)) = {𝑦})
70		eqidd 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑦) = (𝐹‘𝑦))
71		fniniseg 7055	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑦 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹‘𝑦) = (𝐹‘𝑦))))
72	58, 71	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑦 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹‘𝑦) = (𝐹‘𝑦))))
73	43, 70, 72	mpbir2and 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}))
74	73	snssd 4752	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
75	69, 74	eqsstrd 3971	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦 − 𝑥))(AP‘1)(𝑦 − 𝑥)) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
76	62, 75	unssd 4145	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → ({𝑥} ∪ ((𝑥 + (𝑦 − 𝑥))(AP‘1)(𝑦 − 𝑥))) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
77	55, 76	eqsstrd 3971	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦 − 𝑥)) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
78		oveq1 7417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
79	78	sseq1d 3968	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
80		oveq2 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑦 − 𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦 − 𝑥)))
81	80	sseq1d 3968	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑦 − 𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)(𝑦 − 𝑥)) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
82	79, 81	rspc2ev 3594	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℕ ∧ (𝑦 − 𝑥) ∈ ℕ ∧ (𝑥(AP‘2)(𝑦 − 𝑥)) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
83	41, 48, 77, 82	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
84		fvex 6894	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑦) ∈ V
85		sneq 4599	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝐹‘𝑦) → {𝑐} = {(𝐹‘𝑦)})
86	85	imaeq2d 6062	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝐹‘𝑦) → (^◡𝐹 “ {𝑐}) = (^◡𝐹 “ {(𝐹‘𝑦)}))
87	86	sseq2d 3969	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝐹‘𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
88	87	2rexbidv 3230	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝐹‘𝑦) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
89	84, 88	spcev 3565	. . . . . . . . . . . . . 14 ⊢ (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐}))
90	83, 89	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐}))
91		ovex 7443	. . . . . . . . . . . . . 14 ⊢ (1...((♯‘𝑅) + 1)) ∈ V
92		2nn0 12516	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ₀
93	92	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℕ₀)
94	91, 93, 57	vdwmc 17033	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (2 MonoAP 𝐹 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐})))
95	90, 94	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
96	39, 95	sylanl2 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
97	96	expr 461	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑥 < 𝑦 → 2 MonoAP 𝐹))
98	38, 97	mtod 201	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ¬ 𝑥 < 𝑦)
99		simplr1 1234	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 ∈ (1...((♯‘𝑅) + 1)))
100	99, 32	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 ∈ ℝ)
101		simplr2 1235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
102	33, 101	sselid 3935	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ ℝ)
103	100, 102	eqleltd 11349	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 < 𝑦)))
104	36, 98, 103	mpbir2and 725	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
105	104	ex 417	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
106	21, 30, 34, 35, 105	wlogle 11742	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
107	106	ralrimivva 3208	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
108		dff13 7252	. . . . 5 ⊢ (𝐹:(1...((♯‘𝑅) + 1))–1-1→𝑅 ↔ (𝐹:(1...((♯‘𝑅) + 1))⟶𝑅 ∧ ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
109	16, 107, 108	sylanbrc 594	. . . 4 ⊢ (𝜑 → 𝐹:(1...((♯‘𝑅) + 1))–1-1→𝑅)
110		f1domg 8964	. . . 4 ⊢ (𝑅 ∈ Fin → (𝐹:(1...((♯‘𝑅) + 1))–1-1→𝑅 → (1...((♯‘𝑅) + 1)) ≼ 𝑅))
111	1, 109, 110	sylc 66	. . 3 ⊢ (𝜑 → (1...((♯‘𝑅) + 1)) ≼ 𝑅)
112		domnsym 9087	. . 3 ⊢ ((1...((♯‘𝑅) + 1)) ≼ 𝑅 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
113	111, 112	syl 18	. 2 ⊢ (𝜑 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
114	15, 113	pm2.65i 196	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ∪ cun 3903 ⊆ wss 3905 {csn 4589 class class class wbr 5109 ^◡ccnv 5660 “ cima 5664 Fn wfn 6531 ⟶wf 6532 –1-1→wf1 6533 ‘cfv 6536 (class class class)co 7410 ≼ cdom 8937 ≺ csdm 8938 Fincfn 8939 ℝcr 11094 1c1 11096 + caddc 11098 < clt 11238 ≤ cle 11239 − cmin 11436 ℕcn 12228 2c2 12290 ℕ₀cn0 12499 ...cfz 13530 ♯chash 14362 APcvdwa 17020 MonoAP cvdwm 17021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-n0 12500 df-xnn0 12573 df-z 12587 df-uz 12858 df-fz 13531 df-hash 14363 df-vdwap 17023 df-vdwmc 17024

This theorem is referenced by: vdwlem13 17048

W3C validator