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Theorem vdwlem12 16875

Description: Lemma for vdw 16877. 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 14266	. . . . . . 7 ⊢ (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ₀)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝑅) ∈ ℕ₀)
4	3	nn0red 12483	. . . . 5 ⊢ (𝜑 → (♯‘𝑅) ∈ ℝ)
5	4	ltp1d 12094	. . . 4 ⊢ (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
6		nn0p1nn 12461	. . . . . . 7 ⊢ ((♯‘𝑅) ∈ ℕ₀ → ((♯‘𝑅) + 1) ∈ ℕ)
7	3, 6	syl 17	. . . . . 6 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
8	7	nnnn0d 12482	. . . . 5 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ₀)
9		hashfz1 14256	. . . . 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 5138	. . 3 ⊢ (𝜑 → (♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))))
12		fzfi 13887	. . . 4 ⊢ (1...((♯‘𝑅) + 1)) ∈ Fin
13		hashsdom 14291	. . . 4 ⊢ ((𝑅 ∈ Fin ∧ (1...((♯‘𝑅) + 1)) ∈ Fin) → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
14	1, 12, 13	sylancl 586	. . 3 ⊢ (𝜑 → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
15	11, 14	mpbid 231	. 2 ⊢ (𝜑 → 𝑅 ≺ (1...((♯‘𝑅) + 1)))
16		vdwlem12.f	. . . . 5 ⊢ (𝜑 → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
17		fveq2 6847	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
18		fveq2 6847	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦))
19	17, 18	eqeqan12d 2745	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
20		eqeq12 2748	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝑧 = 𝑤 ↔ 𝑥 = 𝑦))
21	19, 20	imbi12d 344	. . . . . . 7 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
22		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
23		fveq2 6847	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
24	22, 23	eqeqan12d 2745	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
25		eqcom 2738	. . . . . . . . 9 ⊢ ((𝐹‘𝑦) = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = (𝐹‘𝑦))
26	24, 25	bitrdi 286	. . . . . . . 8 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
27		eqeq12 2748	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑧 = 𝑤 ↔ 𝑦 = 𝑥))
28		eqcom 2738	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
29	27, 28	bitrdi 286	. . . . . . . 8 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑧 = 𝑤 ↔ 𝑥 = 𝑦))
30	26, 29	imbi12d 344	. . . . . . 7 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
31		elfznn 13480	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℕ)
32	31	nnred 12177	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℝ)
33	32	ssriv 3951	. . . . . . . 8 ⊢ (1...((♯‘𝑅) + 1)) ⊆ ℝ
34	33	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1...((♯‘𝑅) + 1)) ⊆ ℝ)
35		biidd 261	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
36		simplr3 1217	. . . . . . . . 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 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ¬ 2 MonoAP 𝐹)
39		3simpa 1148	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦) → (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1))))
40		simplrl 775	. . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
43		simplrr 776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
44		elfznn 13480	. . . . . . . . . . . . . . . . . 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 12206	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℕ))
47	41, 45, 46	syl2anc 584	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑦 − 𝑥) ∈ ℕ)
49		df-2 12225	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 = (1 + 1)
50	49	fveq2i 6850	. . . . . . . . . . . . . . . . . 18 ⊢ (AP‘2) = (AP‘(1 + 1))
51	50	oveqi 7375	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(AP‘2)(𝑦 − 𝑥)) = (𝑥(AP‘(1 + 1))(𝑦 − 𝑥))
52		1nn0 12438	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
53		vdwapun 16857	. . . . . . . . . . . . . . . . . 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 1466	. . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) = (𝐹‘𝑦))
57	16	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
58	57	ffnd 6674	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((♯‘𝑅) + 1)))
59		fniniseg 7015	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑥 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))))
60	58, 59	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))))
61	40, 56, 60	mpbir2and 711	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}))
62	61	snssd 4774	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
63	41	nncnd 12178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
64	45	nncnd 12178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
65	63, 64	pncan3d 11524	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦 − 𝑥)) = 𝑦)
66	65	oveq1d 7377	. . . . . . . . . . . . . . . . . . 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 16860	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 − 𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦 − 𝑥)) = {𝑦})
68	45, 48, 67	syl2anc 584	. . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑦) = (𝐹‘𝑦))
71		fniniseg 7015	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 Fn (1...((♯‘𝑅) + 1)) → (𝑦 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ (𝐹‘𝑦) = (𝐹‘𝑦))))
72	58, 71	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))))
73	43, 70, 72	mpbir2and 711	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}))
74	73	snssd 4774	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
75	69, 74	eqsstrd 3985	. . . . . . . . . . . . . . . . 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 4151	. . . . . . . . . . . . . . . 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 3985	. . . . . . . . . . . . . . 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 7369	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
79	78	sseq1d 3978	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
80		oveq2 7370	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑦 − 𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦 − 𝑥)))
81	80	sseq1d 3978	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑦 − 𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)(𝑦 − 𝑥)) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
82	79, 81	rspc2ev 3593	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 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 6860	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑦) ∈ V
85		sneq 4601	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝐹‘𝑦) → {𝑐} = {(𝐹‘𝑦)})
86	85	imaeq2d 6018	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝐹‘𝑦) → (^◡𝐹 “ {𝑐}) = (^◡𝐹 “ {(𝐹‘𝑦)}))
87	86	sseq2d 3979	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝐹‘𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
88	87	2rexbidv 3209	. . . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . . 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 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}))
91		ovex 7395	. . . . . . . . . . . . . 14 ⊢ (1...((♯‘𝑅) + 1)) ∈ V
92		2nn0 12439	. . . . . . . . . . . . . . 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 16861	. . . . . . . . . . . . 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 256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
96	39, 95	sylanl2 679	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
97	96	expr 457	. . . . . . . . . 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 197	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → ¬ 𝑥 < 𝑦)
99		simplr1 1215	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 ∈ (1...((♯‘𝑅) + 1)))
100	99, 32	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 ∈ ℝ)
101		simplr2 1216	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
102	33, 101	sselid 3945	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ ℝ)
103	100, 102	eqleltd 11308	. . . . . . . . 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 711	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
105	104	ex 413	. . . . . . 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 11697	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
107	106	ralrimivva 3193	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
108		dff13 7207	. . . . 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 583	. . . 4 ⊢ (𝜑 → 𝐹:(1...((♯‘𝑅) + 1))–1-1→𝑅)
110		f1domg 8919	. . . 4 ⊢ (𝑅 ∈ Fin → (𝐹:(1...((♯‘𝑅) + 1))–1-1→𝑅 → (1...((♯‘𝑅) + 1)) ≼ 𝑅))
111	1, 109, 110	sylc 65	. . 3 ⊢ (𝜑 → (1...((♯‘𝑅) + 1)) ≼ 𝑅)
112		domnsym 9050	. . 3 ⊢ ((1...((♯‘𝑅) + 1)) ≼ 𝑅 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
113	111, 112	syl 17	. 2 ⊢ (𝜑 → ¬ 𝑅 ≺ (1...((♯‘𝑅) + 1)))
114	15, 113	pm2.65i 193	1 ⊢ ¬ 𝜑

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3060 ∃wrex 3069 ∪ cun 3911 ⊆ wss 3913 {csn 4591 class class class wbr 5110 ^◡ccnv 5637 “ cima 5641 Fn wfn 6496 ⟶wf 6497 –1-1→wf1 6498 ‘cfv 6501 (class class class)co 7362 ≼ cdom 8888 ≺ csdm 8889 Fincfn 8890 ℝcr 11059 1c1 11061 + caddc 11063 < clt 11198 ≤ cle 11199 − cmin 11394 ℕcn 12162 2c2 12217 ℕ₀cn0 12422 ...cfz 13434 ♯chash 14240 APcvdwa 16848 MonoAP cvdwm 16849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9884 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-nn 12163 df-2 12225 df-n0 12423 df-xnn0 12495 df-z 12509 df-uz 12773 df-fz 13435 df-hash 14241 df-vdwap 16851 df-vdwmc 16852

This theorem is referenced by: vdwlem13 16876

W3C validator