vdwlem12 - Metamath Proof Explorer

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

Theorem vdwlem12 17179

Description: Lemma for vdw 17181. 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 14507	. . . . . . 7 ⊢ (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ₀)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘𝑅) ∈ ℕ₀)
4	3	nn0red 12675	. . . . 5 ⊢ (𝜑 → (♯‘𝑅) ∈ ℝ)
5	4	ltp1d 12254	. . . 4 ⊢ (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
6		nn0p1nn 12652	. . . . . . 7 ⊢ ((♯‘𝑅) ∈ ℕ₀ → ((♯‘𝑅) + 1) ∈ ℕ)
7	3, 6	syl 18	. . . . . 6 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
8	7	nnnn0d 12674	. . . . 5 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ₀)
9		hashfz1 14497	. . . . 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 5132	. . 3 ⊢ (𝜑 → (♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))))
12		fzfi 14122	. . . 4 ⊢ (1...((♯‘𝑅) + 1)) ∈ Fin
13		hashsdom 14532	. . . 4 ⊢ ((𝑅 ∈ Fin ∧ (1...((♯‘𝑅) + 1)) ∈ Fin) → ((♯‘𝑅) < (♯‘(1...((♯‘𝑅) + 1))) ↔ 𝑅 ≺ (1...((♯‘𝑅) + 1))))
14	1, 12, 13	sylancl 598	. . 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 6890	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
18		fveq2 6890	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦))
19	17, 18	eqeqan12d 2774	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
20		eqeq12 2777	. . . . . . . 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 6890	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
23		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
24	22, 23	eqeqan12d 2774	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
25		eqcom 2767	. . . . . . . . 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 2777	. . . . . . . . 9 ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝑧 = 𝑤 ↔ 𝑦 = 𝑥))
28		eqcom 2767	. . . . . . . . 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 13694	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℕ)
32	31	nnred 12357	. . . . . . . . 9 ⊢ (𝑥 ∈ (1...((♯‘𝑅) + 1)) → 𝑥 ∈ ℝ)
33	32	ssriv 3913	. . . . . . . 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 739	. . . . . . . . . 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 789	. . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
43		simplrr 790	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (1...((♯‘𝑅) + 1)))
44		elfznn 13694	. . . . . . . . . . . . . . . . . 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 12389	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 < 𝑦 ↔ (𝑦 − 𝑥) ∈ ℕ))
47	41, 45, 46	syl2anc 596	. . . . . . . . . . . . . . . 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 12412	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 = (1 + 1)
50	49	fveq2i 6893	. . . . . . . . . . . . . . . . . 18 ⊢ (AP‘2) = (AP‘(1 + 1))
51	50	oveqi 7437	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(AP‘2)(𝑦 − 𝑥)) = (𝑥(AP‘(1 + 1))(𝑦 − 𝑥))
52		1nn0 12629	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
53		vdwapun 17161	. . . . . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) = (𝐹‘𝑦))
57	16	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝐹:(1...((♯‘𝑅) + 1))⟶𝑅)
58	57	ffnd 6715	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝐹 Fn (1...((♯‘𝑅) + 1)))
59		fniniseg 7064	. . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}))
62	61	snssd 4743	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑥} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
63	41	nncnd 12358	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℂ)
64	45	nncnd 12358	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℂ)
65	63, 64	pncan3d 11679	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 + (𝑦 − 𝑥)) = 𝑦)
66	65	oveq1d 7439	. . . . . . . . . . . . . . . . . . 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 17164	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ ∧ (𝑦 − 𝑥) ∈ ℕ) → (𝑦(AP‘1)(𝑦 − 𝑥)) = {𝑦})
68	45, 48, 67	syl2anc 596	. . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑦) = (𝐹‘𝑦))
71		fniniseg 7064	. . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ (^◡𝐹 “ {(𝐹‘𝑦)}))
74	73	snssd 4743	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → {𝑦} ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}))
75	69, 74	eqsstrd 3943	. . . . . . . . . . . . . . . . 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 4189	. . . . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑎(AP‘2)𝑑) = (𝑥(AP‘2)𝑑))
79	78	sseq1d 3940	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
80		oveq2 7432	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑦 − 𝑥) → (𝑥(AP‘2)𝑑) = (𝑥(AP‘2)(𝑦 − 𝑥)))
81	80	sseq1d 3940	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑦 − 𝑥) → ((𝑥(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)}) ↔ (𝑥(AP‘2)(𝑦 − 𝑥)) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
82	79, 81	rspc2ev 3588	. . . . . . . . . . . . . . 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 6903	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑦) ∈ V
85		sneq 4590	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = (𝐹‘𝑦) → {𝑐} = {(𝐹‘𝑦)})
86	85	imaeq2d 6054	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝐹‘𝑦) → (^◡𝐹 “ {𝑐}) = (^◡𝐹 “ {(𝐹‘𝑦)}))
87	86	sseq2d 3941	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝐹‘𝑦) → ((𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {𝑐}) ↔ (𝑎(AP‘2)𝑑) ⊆ (^◡𝐹 “ {(𝐹‘𝑦)})))
88	87	2rexbidv 3227	. . . . . . . . . . . . . . 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 3560	. . . . . . . . . . . . . 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 7457	. . . . . . . . . . . . . 14 ⊢ (1...((♯‘𝑅) + 1)) ∈ V
92		2nn0 12630	. . . . . . . . . . . . . . 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 17165	. . . . . . . . . . . . 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 694	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦)) → 2 MonoAP 𝐹)
97	96	expr 462	. . . . . . . . . 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 3907	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑦 ∈ ℝ)
103	100, 102	eqleltd 11461	. . . . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑥 ≤ 𝑦)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
105	104	ex 418	. . . . . . 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 11856	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...((♯‘𝑅) + 1)) ∧ 𝑦 ∈ (1...((♯‘𝑅) + 1)))) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
107	106	ralrimivva 3205	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (1...((♯‘𝑅) + 1))∀𝑦 ∈ (1...((♯‘𝑅) + 1))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
108		dff13 7262	. . . . 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 595	. . . 4 ⊢ (𝜑 → 𝐹:(1...((♯‘𝑅) + 1))–1-1→𝑅)
110		f1domg 9005	. . . 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 9129	. . 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∃wex 1811 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 ⊆ wss 3896 ∪ cun 4148 {csn 4584 class class class wbr 5102 ^◡ccnv 5650 “ cima 5654 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7424 ≼ cdom 8978 ≺ csdm 8979 Fincfn 8980 ℝcr 11206 1c1 11208 + caddc 11210 < clt 11350 ≤ cle 11351 − cmin 11548 ℕcn 12342 2c2 12404 ℕ₀cn0 12613 ...cfz 13646 ♯chash 14481 APcvdwa 17152 MonoAP cvdwm 17153

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-oadd 8487 df-er 8724 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-2 12412 df-n0 12614 df-xnn0 12687 df-z 12701 df-uz 12973 df-fz 13647 df-hash 14482 df-vdwap 17155 df-vdwmc 17156

This theorem is used by: vdwlem13 17180

W3C validator