subfacp1lem1 - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > subfacp1lem1		Structured version Visualization version GIF version

Theorem subfacp1lem1 36152

Description: Lemma for subfacp1 36159. The set 𝐾 together with {1, 𝑀} partitions the set 1...(𝑁 + 1). (Contributed by Mario Carneiro, 23-Jan-2015.)

**Hypotheses**
Ref	Expression
derang.d	⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥⟶1-1onto𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
subfac.n	⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ (𝐷‘(1...𝑛)))
subfacp1lem.a	⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))⟶1-1onto(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)}
subfacp1lem1.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
subfacp1lem1.m	⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))
subfacp1lem1.x	⊢ 𝑀 ∈ V
subfacp1lem1.k	⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})

**Assertion**
Ref	Expression
subfacp1lem1	⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1)))

Distinct variable groups: 𝑓,𝑛,𝑥,𝑦,𝐴 𝑓,𝑁,𝑛,𝑥,𝑦 𝜑,𝑥,𝑦 𝐷,𝑛 𝑓,𝐾,𝑛,𝑥,𝑦 𝑓,𝑀,𝑥,𝑦 𝑆,𝑛,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑓, 𝑛) 𝐷(𝑥, 𝑦, 𝑓) 𝑆(𝑓) 𝑀(𝑛)

**Proof of Theorem subfacp1lem1**
Step	Hyp	Ref	Expression
1		disj 4406	. . . 4 ⊢ ((𝐾 ∩ {1, 𝑀}) = ∅ ↔ ∀𝑥 ∈ 𝐾 ¬ 𝑥 ∈ {1, 𝑀})
2		eldifi 4070	. . . . . . . . 9 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → 𝑥 ∈ (2...(𝑁 + 1)))
3		elfzle1 13700	. . . . . . . . 9 ⊢ (𝑥 ∈ (2...(𝑁 + 1)) → 2 ≤ 𝑥)
4		1lt2 12555	. . . . . . . . . . . 12 ⊢ 1 < 2
5		1re 11347	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
6		2re 12457	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
7	5, 6	ltnlei 11470	. . . . . . . . . . . 12 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
8	4, 7	mpbi 233	. . . . . . . . . . 11 ⊢ ¬ 2 ≤ 1
9		breq2 5112	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (2 ≤ 𝑥 ↔ 2 ≤ 1))
10	8, 9	mtbiri 330	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ¬ 2 ≤ 𝑥)
11	10	necon2ai 2988	. . . . . . . . 9 ⊢ (2 ≤ 𝑥 → 𝑥 ≠ 1)
12	2, 3, 11	3syl 19	. . . . . . . 8 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → 𝑥 ≠ 1)
13		eldifsni 4755	. . . . . . . 8 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → 𝑥 ≠ 𝑀)
14	12, 13	jca 521	. . . . . . 7 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → (𝑥 ≠ 1 ∧ 𝑥 ≠ 𝑀))
15		subfacp1lem1.k	. . . . . . 7 ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})
16	14, 15	eleq2s 2882	. . . . . 6 ⊢ (𝑥 ∈ 𝐾 → (𝑥 ≠ 1 ∧ 𝑥 ≠ 𝑀))
17		neanior 3052	. . . . . 6 ⊢ ((𝑥 ≠ 1 ∧ 𝑥 ≠ 𝑀) ↔ ¬ (𝑥 = 1 ∨ 𝑥 = 𝑀))
18	16, 17	sylib 221	. . . . 5 ⊢ (𝑥 ∈ 𝐾 → ¬ (𝑥 = 1 ∨ 𝑥 = 𝑀))
19		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
20	19	elpr 4609	. . . . 5 ⊢ (𝑥 ∈ {1, 𝑀} ↔ (𝑥 = 1 ∨ 𝑥 = 𝑀))
21	18, 20	sylnibr 332	. . . 4 ⊢ (𝑥 ∈ 𝐾 → ¬ 𝑥 ∈ {1, 𝑀})
22	1, 21	mprgbir 3087	. . 3 ⊢ (𝐾 ∩ {1, 𝑀}) = ∅
23	22	a1i 11	. 2 ⊢ (𝜑 → (𝐾 ∩ {1, 𝑀}) = ∅)
24		uncom 4160	. . . 4 ⊢ ({1} ∪ (𝐾 ∪ {𝑀})) = ((𝐾 ∪ {𝑀}) ∪ {1})
25		1z 12766	. . . . . 6 ⊢ 1 ∈ ℤ
26		fzsn 13740	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
27	25, 26	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
28	15	uneq1i 4166	. . . . . 6 ⊢ (𝐾 ∪ {𝑀}) = (((2...(𝑁 + 1)) ∖ {𝑀}) ∪ {𝑀})
29		undif1 4433	. . . . . 6 ⊢ (((2...(𝑁 + 1)) ∖ {𝑀}) ∪ {𝑀}) = ((2...(𝑁 + 1)) ∪ {𝑀})
30	28, 29	eqtr2i 2787	. . . . 5 ⊢ ((2...(𝑁 + 1)) ∪ {𝑀}) = (𝐾 ∪ {𝑀})
31	27, 30	uneq12i 4168	. . . 4 ⊢ ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = ({1} ∪ (𝐾 ∪ {𝑀}))
32		df-pr 4591	. . . . . . 7 ⊢ {1, 𝑀} = ({1} ∪ {𝑀})
33	32	equncomi 4162	. . . . . 6 ⊢ {1, 𝑀} = ({𝑀} ∪ {1})
34	33	uneq2i 4167	. . . . 5 ⊢ (𝐾 ∪ {1, 𝑀}) = (𝐾 ∪ ({𝑀} ∪ {1}))
35		unass 4173	. . . . 5 ⊢ ((𝐾 ∪ {𝑀}) ∪ {1}) = (𝐾 ∪ ({𝑀} ∪ {1}))
36	34, 35	eqtr4i 2789	. . . 4 ⊢ (𝐾 ∪ {1, 𝑀}) = ((𝐾 ∪ {𝑀}) ∪ {1})
37	24, 31, 36	3eqtr4i 2796	. . 3 ⊢ ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = (𝐾 ∪ {1, 𝑀})
38		subfacp1lem1.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))
39	38	snssd 4749	. . . . . . 7 ⊢ (𝜑 → {𝑀} ⊆ (2...(𝑁 + 1)))
40		ssequn2 4190	. . . . . . 7 ⊢ ({𝑀} ⊆ (2...(𝑁 + 1)) ↔ ((2...(𝑁 + 1)) ∪ {𝑀}) = (2...(𝑁 + 1)))
41	39, 40	sylib 221	. . . . . 6 ⊢ (𝜑 → ((2...(𝑁 + 1)) ∪ {𝑀}) = (2...(𝑁 + 1)))
42		df-2 12445	. . . . . . 7 ⊢ 2 = (1 + 1)
43	42	oveq1i 7461	. . . . . 6 ⊢ (2...(𝑁 + 1)) = ((1 + 1)...(𝑁 + 1))
44	41, 43	eqtrdi 2814	. . . . 5 ⊢ (𝜑 → ((2...(𝑁 + 1)) ∪ {𝑀}) = ((1 + 1)...(𝑁 + 1)))
45	44	uneq2d 4170	. . . 4 ⊢ (𝜑 → ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = ((1...1) ∪ ((1 + 1)...(𝑁 + 1))))
46		subfacp1lem1.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47	46	peano2nnd 12392	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
48		nnuz 13044	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
49	47, 48	eleqtrdi 2874	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘1))
50		eluzfz1 13704	. . . . 5 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑁 + 1)))
51		fzsplit 13724	. . . . 5 ⊢ (1 ∈ (1...(𝑁 + 1)) → (1...(𝑁 + 1)) = ((1...1) ∪ ((1 + 1)...(𝑁 + 1))))
52	49, 50, 51	3syl 19	. . . 4 ⊢ (𝜑 → (1...(𝑁 + 1)) = ((1...1) ∪ ((1 + 1)...(𝑁 + 1))))
53	45, 52	eqtr4d 2801	. . 3 ⊢ (𝜑 → ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = (1...(𝑁 + 1)))
54	37, 53	eqtr3id 2812	. 2 ⊢ (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
55	42	oveq2i 7462	. . 3 ⊢ ((𝑁 + 1) − 2) = ((𝑁 + 1) − (1 + 1))
56		fzfi 14155	. . . . . . . . 9 ⊢ (2...(𝑁 + 1)) ∈ Fin
57		diffi 9228	. . . . . . . . 9 ⊢ ((2...(𝑁 + 1)) ∈ Fin → ((2...(𝑁 + 1)) ∖ {𝑀}) ∈ Fin)
58	56, 57	ax-mp 5	. . . . . . . 8 ⊢ ((2...(𝑁 + 1)) ∖ {𝑀}) ∈ Fin
59	15, 58	eqeltri 2860	. . . . . . 7 ⊢ 𝐾 ∈ Fin
60		prfi 9353	. . . . . . 7 ⊢ {1, 𝑀} ∈ Fin
61		hashun 14567	. . . . . . 7 ⊢ ((𝐾 ∈ Fin ∧ {1, 𝑀} ∈ Fin ∧ (𝐾 ∩ {1, 𝑀}) = ∅) → (♯‘(𝐾 ∪ {1, 𝑀})) = ((♯‘𝐾) + (♯‘{1, 𝑀})))
62	59, 60, 22, 61	mp3an 1491	. . . . . 6 ⊢ (♯‘(𝐾 ∪ {1, 𝑀})) = ((♯‘𝐾) + (♯‘{1, 𝑀}))
63	54	fveq2d 6920	. . . . . 6 ⊢ (𝜑 → (♯‘(𝐾 ∪ {1, 𝑀})) = (♯‘(1...(𝑁 + 1))))
64		neeq1 3021	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑥 ≠ 1 ↔ 𝑀 ≠ 1))
65	3, 11	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2...(𝑁 + 1)) → 𝑥 ≠ 1)
66	64, 65	vtoclga 3540	. . . . . . . . . 10 ⊢ (𝑀 ∈ (2...(𝑁 + 1)) → 𝑀 ≠ 1)
67	38, 66	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 1)
68	67	necomd 3014	. . . . . . . 8 ⊢ (𝜑 → 1 ≠ 𝑀)
69		1ex 11342	. . . . . . . . 9 ⊢ 1 ∈ V
70		subfacp1lem1.x	. . . . . . . . 9 ⊢ 𝑀 ∈ V
71		hashprg 14580	. . . . . . . . 9 ⊢ ((1 ∈ V ∧ 𝑀 ∈ V) → (1 ≠ 𝑀 ↔ (♯‘{1, 𝑀}) = 2))
72	69, 70, 71	mp2an 705	. . . . . . . 8 ⊢ (1 ≠ 𝑀 ↔ (♯‘{1, 𝑀}) = 2)
73	68, 72	sylib 221	. . . . . . 7 ⊢ (𝜑 → (♯‘{1, 𝑀}) = 2)
74	73	oveq2d 7467	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐾) + (♯‘{1, 𝑀})) = ((♯‘𝐾) + 2))
75	62, 63, 74	3eqtr3a 2822	. . . . 5 ⊢ (𝜑 → (♯‘(1...(𝑁 + 1))) = ((♯‘𝐾) + 2))
76	47	nnnn0d 12707	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
77		hashfz1 14531	. . . . . 6 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
78	76, 77	syl 18	. . . . 5 ⊢ (𝜑 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
79	75, 78	eqtr3d 2800	. . . 4 ⊢ (𝜑 → ((♯‘𝐾) + 2) = (𝑁 + 1))
80	47	nncnd 12391	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
81		2cnd 12461	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
82		hashcl 14541	. . . . . . . 8 ⊢ (𝐾 ∈ Fin → (♯‘𝐾) ∈ ℕ₀)
83	59, 82	ax-mp 5	. . . . . . 7 ⊢ (♯‘𝐾) ∈ ℕ₀
84	83	nn0cni 12658	. . . . . 6 ⊢ (♯‘𝐾) ∈ ℂ
85	84	a1i 11	. . . . 5 ⊢ (𝜑 → (♯‘𝐾) ∈ ℂ)
86	80, 81, 85	subadd2d 11727	. . . 4 ⊢ (𝜑 → (((𝑁 + 1) − 2) = (♯‘𝐾) ↔ ((♯‘𝐾) + 2) = (𝑁 + 1)))
87	79, 86	mpbird 260	. . 3 ⊢ (𝜑 → ((𝑁 + 1) − 2) = (♯‘𝐾))
88	46	nncnd 12391	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
89		1cnd 11341	. . . 4 ⊢ (𝜑 → 1 ∈ ℂ)
90	88, 89, 89	pnpcan2d 11746	. . 3 ⊢ (𝜑 → ((𝑁 + 1) − (1 + 1)) = (𝑁 − 1))
91	55, 87, 90	3eqtr3a 2822	. 2 ⊢ (𝜑 → (♯‘𝐾) = (𝑁 − 1))
92	23, 54, 91	3jca 1146	1 ⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 {cab 2741 ≠ wne 2959 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 {csn 4588 {cpr 4590 class class class wbr 5108 ↦ cmpt 5192 ⟶1-1ontowf1o 6565 ‘cfv 6566 (class class class)co 7451 Fincfn 9011 ℂcc 11237 1c1 11240 + caddc 11242 < clt 11382 ≤ cle 11383 − cmin 11580 ℕcn 12375 2c2 12437 ℕ₀cn0 12646 ℤcz 12733 ℤ_≥cuz 13005 ...cfz 13679 ♯chash 14515

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-n0 12647 df-z 12734 df-uz 13006 df-fz 13680 df-hash 14516

This theorem is used by: subfacp1lem2a 36153 subfacp1lem3 36155 subfacp1lem4 36156

W3C validator