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Theorem vdwlem12 16987

Description: Lemma for vdw 16989. 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 14366	. . . . . . 7 ⊢ (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ₀)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘𝑅) ∈ ℕ₀)
4	3	nn0red 12577	. . . . 5 ⊢ (𝜑 → (♯‘𝑅) ∈ ℝ)
5	4	ltp1d 12188	. . . 4 ⊢ (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
6		nn0p1nn 12555	. . . . . . 7 ⊢ ((♯‘𝑅) ∈ ℕ₀ → ((♯‘𝑅) + 1) ∈ ℕ)
7	3, 6	syl 17	. . . . . 6 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
8	7	nnnn0d 12576	. . . . 5 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ₀)
9		hashfz1 14356	. . . . 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 5172	. . 3 ⊢ (𝜑 → (♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))))
12		fzfi 13984	. . . 4 ⊢ (1...((♯‘𝑅) + 1)) ∈ Fin
13		hashsdom 14391	. . . 4 ⊢ ((𝑅 ∈ Fin ∧ (1...((♯‘𝑅) + 1)) ∈ Fin) → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
14	1, 12, 13	sylancl 584	. . 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 6891	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
18		fveq2 6891	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦))
19	17, 18	eqeqan12d 2740	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
20		eqeq12 2743	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝑧 = 𝑤 ↔ 𝑥 = 𝑦))
21	19, 20	imbi12d 343	. . . . . . 7 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
22		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
23		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
24	22, 23	eqeqan12d 2740	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
25		eqcom 2733	. . . . . . . . 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 2743	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑧 = 𝑤 ↔ 𝑦 = 𝑥))
28		eqcom 2733	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
29	27, 28	bitrdi 286	. . . . . . . 8 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑧 = 𝑤 ↔ 𝑥 = 𝑦))
30	26, 29	imbi12d 343	. . . . . . 7 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
31		elfznn 13576	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℕ)
32	31	nnred 12271	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℝ)
33	32	ssriv 3983	. . . . . . . 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 1214	. . . . . . . . 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 1145	. . . . . . . . . . . 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 13576	. . . . . . . . . . . . . . . . . 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 12300	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℕ))
47	41, 45, 46	syl2anc 582	. . . . . . . . . . . . . . . 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 12319	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 = (1 + 1)
50	49	fveq2i 6894	. . . . . . . . . . . . . . . . . 18 ⊢ (AP‘2) = (AP‘(1 + 1))
51	50	oveqi 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(AP‘2)(𝑦 − 𝑥)) = (𝑥(AP‘(1 + 1))(𝑦 − 𝑥))
52		1nn0 12532	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
53		vdwapun 16969	. . . . . . . . . . . . . . . . . 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 1463	. . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . 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 6719	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((♯‘𝑅) + 1)))
59		fniniseg 7063	. . . . . . . . . . . . . . . . . . . 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 4809	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
63	41	nncnd 12272	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
64	45	nncnd 12272	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
65	63, 64	pncan3d 11613	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦 − 𝑥)) = 𝑦)
66	65	oveq1d 7429	. . . . . . . . . . . . . . . . . . 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 16972	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 − 𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦 − 𝑥)) = {𝑦})
68	45, 48, 67	syl2anc 582	. . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑦) = (𝐹‘𝑦))
71		fniniseg 7063	. . . . . . . . . . . . . . . . . . . . 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 4809	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
75	69, 74	eqsstrd 4018	. . . . . . . . . . . . . . . . 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 4185	. . . . . . . . . . . . . . . 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 4018	. . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
79	78	sseq1d 4011	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
80		oveq2 7422	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑦 − 𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦 − 𝑥)))
81	80	sseq1d 4011	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑦 − 𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)(𝑦 − 𝑥)) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
82	79, 81	rspc2ev 3621	. . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . 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 6904	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑦) ∈ V
85		sneq 4634	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝐹‘𝑦) → {𝑐} = {(𝐹‘𝑦)})
86	85	imaeq2d 6060	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝐹‘𝑦) → (^◡𝐹 “ {𝑐}) = (^◡𝐹 “ {(𝐹‘𝑦)}))
87	86	sseq2d 4012	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝐹‘𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
88	87	2rexbidv 3210	. . . . . . . . . . . . . . 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 3592	. . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . 14 ⊢ (1...((♯‘𝑅) + 1)) ∈ V
92		2nn0 12533	. . . . . . . . . . . . . . 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 16973	. . . . . . . . . . . . 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 455	. . . . . . . . . 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 1212	. . . . . . . . . . 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 1213	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
102	33, 101	sselid 3977	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ ℝ)
103	100, 102	eqleltd 11397	. . . . . . . . 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 411	. . . . . . 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 11786	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
107	106	ralrimivva 3191	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
108		dff13 7260	. . . . 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 581	. . . 4 ⊢ (𝜑 → 𝐹:(1...((♯‘𝑅) + 1))–1-1→𝑅)
110		f1domg 8993	. . . 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 9127	. . 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 394 ∧ w3a 1084 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 ∪ cun 3945 ⊆ wss 3947 {csn 4624 class class class wbr 5144 ^◡ccnv 5672 “ cima 5676 Fn wfn 6539 ⟶wf 6540 –1-1→wf1 6541 ‘cfv 6544 (class class class)co 7414 ≼ cdom 8962 ≺ csdm 8963 Fincfn 8964 ℝcr 11146 1c1 11148 + caddc 11150 < clt 11287 ≤ cle 11288 − cmin 11483 ℕcn 12256 2c2 12311 ℕ₀cn0 12516 ...cfz 13530 ♯chash 14340 APcvdwa 16960 MonoAP cvdwm 16961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-n0 12517 df-xnn0 12589 df-z 12603 df-uz 12867 df-fz 13531 df-hash 14341 df-vdwap 16963 df-vdwmc 16964

This theorem is referenced by: vdwlem13 16988

W3C validator