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Theorem vdwlem12 15534

Description: Lemma for vdw 15536. 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 13009	. . . . . . 7 ⊢ (𝑅 ∈ Fin → (#‘𝑅) ∈ ℕ₀)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘𝑅) ∈ ℕ₀)
4	3	nn0red 11229	. . . . 5 ⊢ (𝜑 → (#‘𝑅) ∈ ℝ)
5	4	ltp1d 10833	. . . 4 ⊢ (𝜑 → (#‘𝑅) < ((#‘𝑅) + 1))
6		nn0p1nn 11209	. . . . . . 7 ⊢ ((#‘𝑅) ∈ ℕ₀ → ((#‘𝑅) + 1) ∈ ℕ)
7	3, 6	syl 17	. . . . . 6 ⊢ (𝜑 → ((#‘𝑅) + 1) ∈ ℕ)
8	7	nnnn0d 11228	. . . . 5 ⊢ (𝜑 → ((#‘𝑅) + 1) ∈ ℕ₀)
9		hashfz1 12996	. . . . 5 ⊢ (((#‘𝑅) + 1) ∈ ℕ₀ → (#‘(1...((#‘𝑅) + 1))) = ((#‘𝑅) + 1))
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → (#‘(1...((#‘𝑅) + 1))) = ((#‘𝑅) + 1))
11	5, 10	breqtrrd 4611	. . 3 ⊢ (𝜑 → (#‘𝑅) < (#‘(1...((#‘𝑅) + 1))))
12		fzfi 12633	. . . 4 ⊢ (1...((#‘𝑅) + 1)) ∈ Fin
13		hashsdom 13031	. . . 4 ⊢ ((𝑅 ∈ Fin ∧ (1...((#‘𝑅) + 1)) ∈ Fin) → ((#‘𝑅) < (#‘(1...((#‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((#‘𝑅) + 1))))
14	1, 12, 13	sylancl 693	. . 3 ⊢ (𝜑 → ((#‘𝑅) < (#‘(1...((#‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((#‘𝑅) + 1))))
15	11, 14	mpbid 221	. 2 ⊢ (𝜑 → 𝑅 ≺ (1...((#‘𝑅) + 1)))
16		vdwlem12.f	. . . . 5 ⊢ (𝜑 → 𝐹:(1...((#‘𝑅) + 1))⟶𝑅)
17		fveq2 6103	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
18		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦))
19	17, 18	eqeqan12d 2626	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
20		eqeq12 2623	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝑧 = 𝑤 ↔ 𝑥 = 𝑦))
21	19, 20	imbi12d 333	. . . . . . 7 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
22		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
23		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
24	22, 23	eqeqan12d 2626	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
25		eqcom 2617	. . . . . . . . 9 ⊢ ((𝐹‘𝑦) = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = (𝐹‘𝑦))
26	24, 25	syl6bb 275	. . . . . . . 8 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
27		eqeq12 2623	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑧 = 𝑤 ↔ 𝑦 = 𝑥))
28		eqcom 2617	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
29	27, 28	syl6bb 275	. . . . . . . 8 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑧 = 𝑤 ↔ 𝑥 = 𝑦))
30	26, 29	imbi12d 333	. . . . . . 7 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
31		elfznn 12241	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((#‘𝑅) + 1)) → 𝑥 ∈ ℕ)
32	31	nnred 10912	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...((#‘𝑅) + 1)) → 𝑥 ∈ ℝ)
33	32	ssriv 3572	. . . . . . . 8 ⊢ (1...((#‘𝑅) + 1)) ⊆ ℝ
34	33	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1...((#‘𝑅) + 1)) ⊆ ℝ)
35		biidd 251	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
36		simplr3 1098	. . . . . . . . 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 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ¬ 2 MonoAP 𝐹)
39		3simpa 1051	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦) → (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1))))
40		simplrl 796	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (1...((#‘𝑅) + 1)))
41	40, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℕ)
42		simprr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
43		simplrr 797	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (1...((#‘𝑅) + 1)))
44		elfznn 12241	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1...((#‘𝑅) + 1)) → 𝑦 ∈ ℕ)
45	43, 44	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℕ)
46		nnsub 10936	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℕ))
47	41, 45, 46	syl2anc 691	. . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑦 − 𝑥) ∈ ℕ)
49		df-2 10956	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 = (1 + 1)
50	49	fveq2i 6106	. . . . . . . . . . . . . . . . . 18 ⊢ (AP‘2) = (AP‘(1 + 1))
51	50	oveqi 6562	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(AP‘2)(𝑦 − 𝑥)) = (𝑥(AP‘(1 + 1))(𝑦 − 𝑥))
52		1nn0 11185	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ₀
53	52	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 1 ∈ ℕ₀)
54		vdwapun 15516	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℕ₀ ∧ 𝑥 ∈ ℕ ∧ (𝑦 − 𝑥) ∈ ℕ) → (𝑥(AP‘(1 + 1))(𝑦 − 𝑥)) = ({𝑥} ∪ ((𝑥 + (𝑦 − 𝑥))(AP‘1)(𝑦 − 𝑥))))
55	53, 41, 48, 54	syl3anc 1318	. . . . . . . . . . . . . . . . 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))))
56	51, 55	syl5eq 2656	. . . . . . . . . . . . . . . 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))))
57		simprl 790	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) = (𝐹‘𝑦))
58	16	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝐹:(1...((#‘𝑅) + 1))⟶𝑅)
59		ffn 5958	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹:(1...((#‘𝑅) + 1))⟶𝑅 → 𝐹 Fn (1...((#‘𝑅) + 1)))
60	58, 59	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((#‘𝑅) + 1)))
61		fniniseg 6246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 Fn (1...((#‘𝑅) + 1)) → (𝑥 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))))
62	60, 61	syl 17	. . . . . . . . . . . . . . . . . . 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))))
63	40, 57, 62	mpbir2and 959	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}))
64	63	snssd 4281	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
65	41	nncnd 10913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
66	45	nncnd 10913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
67	65, 66	pncan3d 10274	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦 − 𝑥)) = 𝑦)
68	67	oveq1d 6564	. . . . . . . . . . . . . . . . . . 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)))
69		vdwap1 15519	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 − 𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦 − 𝑥)) = {𝑦})
70	45, 48, 69	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑦(AP‘1)(𝑦 − 𝑥)) = {𝑦})
71	68, 70	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → ((𝑥 + (𝑦 − 𝑥))(AP‘1)(𝑦 − 𝑥)) = {𝑦})
72		eqidd 2611	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑦) = (𝐹‘𝑦))
73		fniniseg 6246	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn (1...((#‘𝑅) + 1)) → (𝑦 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ (𝐹‘𝑦) = (𝐹‘𝑦))))
74	60, 73	syl 17	. . . . . . . . . . . . . . . . . . . 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))))
75	43, 72, 74	mpbir2and 959	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}))
76	75	snssd 4281	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
77	71, 76	eqsstrd 3602	. . . . . . . . . . . . . . . . 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)}))
78	64, 77	unssd 3751	. . . . . . . . . . . . . . . 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)}))
79	56, 78	eqsstrd 3602	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥(AP‘2)(𝑦 − 𝑥)) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
80		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
81	80	sseq1d 3595	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
82		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑦 − 𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦 − 𝑥)))
83	82	sseq1d 3595	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑦 − 𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)(𝑦 − 𝑥)) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
84	81, 83	rspc2ev 3295	. . . . . . . . . . . . . . 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)}))
85	41, 48, 79, 84	syl3anc 1318	. . . . . . . . . . . . . 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)}))
86		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑦) ∈ V
87		sneq 4135	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝐹‘𝑦) → {𝑐} = {(𝐹‘𝑦)})
88	87	imaeq2d 5385	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝐹‘𝑦) → (^◡𝐹 “ {𝑐}) = (^◡𝐹 “ {(𝐹‘𝑦)}))
89	88	sseq2d 3596	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝐹‘𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
90	89	2rexbidv 3039	. . . . . . . . . . . . . . 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)})))
91	86, 90	spcev 3273	. . . . . . . . . . . . . 14 ⊢ (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐}))
92	85, 91	syl 17	. . . . . . . . . . . . 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}))
93		ovex 6577	. . . . . . . . . . . . . 14 ⊢ (1...((#‘𝑅) + 1)) ∈ V
94		2nn0 11186	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ₀
95	94	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℕ₀)
96	93, 95, 58	vdwmc 15520	. . . . . . . . . . . . 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})))
97	92, 96	mpbird 246	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
98	39, 97	sylanl2 681	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
99	98	expr 641	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑥 < 𝑦 → 2 MonoAP 𝐹))
100	38, 99	mtod 188	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ¬ 𝑥 < 𝑦)
101		simplr1 1096	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 ∈ (1...((#‘𝑅) + 1)))
102	101, 32	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 ∈ ℝ)
103		simplr2 1097	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ (1...((#‘𝑅) + 1)))
104	33, 103	sseldi 3566	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ ℝ)
105	102, 104	eqleltd 10060	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑥 = 𝑦 ↔ (𝑥 ≤ 𝑦 ∧ ¬ 𝑥 < 𝑦)))
106	36, 100, 105	mpbir2and 959	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
107	106	ex 449	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
108	21, 30, 34, 35, 107	wlogle 10440	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((#‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((#‘𝑅) + 1)))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
109	108	ralrimivva 2954	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (1...((#‘𝑅) + 1))∀𝑦 ∈ (1...((#‘𝑅) + 1))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
110		dff13 6416	. . . . 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)))
111	16, 109, 110	sylanbrc 695	. . . 4 ⊢ (𝜑 → 𝐹:(1...((#‘𝑅) + 1))–1-1→𝑅)
112		f1domg 7861	. . . 4 ⊢ (𝑅 ∈ Fin → (𝐹:(1...((#‘𝑅) + 1))–1-1→𝑅 → (1...((#‘𝑅) + 1)) ≼ 𝑅))
113	1, 111, 112	sylc 63	. . 3 ⊢ (𝜑 → (1...((#‘𝑅) + 1)) ≼ 𝑅)
114		domnsym 7971	. . 3 ⊢ ((1...((#‘𝑅) + 1)) ≼ 𝑅 → ¬ 𝑅 ≺ (1...((#‘𝑅) + 1)))
115	113, 114	syl 17	. 2 ⊢ (𝜑 → ¬ 𝑅 ≺ (1...((#‘𝑅) + 1)))
116	15, 115	pm2.65i 184	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∪ cun 3538 ⊆ wss 3540 {csn 4125 class class class wbr 4583 ^◡ccnv 5037 “ cima 5041 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 ≼ cdom 7839 ≺ csdm 7840 Fincfn 7841 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ...cfz 12197 #chash 12979 APcvdwa 15507 MonoAP cvdwm 15508

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-vdwap 15510 df-vdwmc 15511

This theorem is referenced by: vdwlem13 15535

W3C validator