subfacp1lem1 - Mathbox for Mario Carneiro

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Theorem subfacp1lem1 35166

Description: Lemma for subfacp1 35173. 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–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
subfac.n	⊢ 𝑆 = (𝑛 ∈ ℕ₀ ↦ (𝐷‘(1...𝑛)))
subfacp1lem.a	⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(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 4413	. . . 4 ⊢ ((𝐾 ∩ {1, 𝑀}) = ∅ ↔ ∀𝑥 ∈ 𝐾 ¬ 𝑥 ∈ {1, 𝑀})
2		eldifi 4094	. . . . . . . . 9 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → 𝑥 ∈ (2...(𝑁 + 1)))
3		elfzle1 13488	. . . . . . . . 9 ⊢ (𝑥 ∈ (2...(𝑁 + 1)) → 2 ≤ 𝑥)
4		1lt2 12352	. . . . . . . . . . . 12 ⊢ 1 < 2
5		1re 11174	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
6		2re 12260	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
7	5, 6	ltnlei 11295	. . . . . . . . . . . 12 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
8	4, 7	mpbi 230	. . . . . . . . . . 11 ⊢ ¬ 2 ≤ 1
9		breq2 5111	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (2 ≤ 𝑥 ↔ 2 ≤ 1))
10	8, 9	mtbiri 327	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ¬ 2 ≤ 𝑥)
11	10	necon2ai 2954	. . . . . . . . 9 ⊢ (2 ≤ 𝑥 → 𝑥 ≠ 1)
12	2, 3, 11	3syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → 𝑥 ≠ 1)
13		eldifsni 4754	. . . . . . . 8 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → 𝑥 ≠ 𝑀)
14	12, 13	jca 511	. . . . . . 7 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → (𝑥 ≠ 1 ∧ 𝑥 ≠ 𝑀))
15		subfacp1lem1.k	. . . . . . 7 ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})
16	14, 15	eleq2s 2846	. . . . . 6 ⊢ (𝑥 ∈ 𝐾 → (𝑥 ≠ 1 ∧ 𝑥 ≠ 𝑀))
17		neanior 3018	. . . . . 6 ⊢ ((𝑥 ≠ 1 ∧ 𝑥 ≠ 𝑀) ↔ ¬ (𝑥 = 1 ∨ 𝑥 = 𝑀))
18	16, 17	sylib 218	. . . . 5 ⊢ (𝑥 ∈ 𝐾 → ¬ (𝑥 = 1 ∨ 𝑥 = 𝑀))
19		vex 3451	. . . . . 6 ⊢ 𝑥 ∈ V
20	19	elpr 4614	. . . . 5 ⊢ (𝑥 ∈ {1, 𝑀} ↔ (𝑥 = 1 ∨ 𝑥 = 𝑀))
21	18, 20	sylnibr 329	. . . 4 ⊢ (𝑥 ∈ 𝐾 → ¬ 𝑥 ∈ {1, 𝑀})
22	1, 21	mprgbir 3051	. . 3 ⊢ (𝐾 ∩ {1, 𝑀}) = ∅
23	22	a1i 11	. 2 ⊢ (𝜑 → (𝐾 ∩ {1, 𝑀}) = ∅)
24		uncom 4121	. . . 4 ⊢ ({1} ∪ (𝐾 ∪ {𝑀})) = ((𝐾 ∪ {𝑀}) ∪ {1})
25		1z 12563	. . . . . 6 ⊢ 1 ∈ ℤ
26		fzsn 13527	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
27	25, 26	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
28	15	uneq1i 4127	. . . . . 6 ⊢ (𝐾 ∪ {𝑀}) = (((2...(𝑁 + 1)) ∖ {𝑀}) ∪ {𝑀})
29		undif1 4439	. . . . . 6 ⊢ (((2...(𝑁 + 1)) ∖ {𝑀}) ∪ {𝑀}) = ((2...(𝑁 + 1)) ∪ {𝑀})
30	28, 29	eqtr2i 2753	. . . . 5 ⊢ ((2...(𝑁 + 1)) ∪ {𝑀}) = (𝐾 ∪ {𝑀})
31	27, 30	uneq12i 4129	. . . 4 ⊢ ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = ({1} ∪ (𝐾 ∪ {𝑀}))
32		df-pr 4592	. . . . . . 7 ⊢ {1, 𝑀} = ({1} ∪ {𝑀})
33	32	equncomi 4123	. . . . . 6 ⊢ {1, 𝑀} = ({𝑀} ∪ {1})
34	33	uneq2i 4128	. . . . 5 ⊢ (𝐾 ∪ {1, 𝑀}) = (𝐾 ∪ ({𝑀} ∪ {1}))
35		unass 4135	. . . . 5 ⊢ ((𝐾 ∪ {𝑀}) ∪ {1}) = (𝐾 ∪ ({𝑀} ∪ {1}))
36	34, 35	eqtr4i 2755	. . . 4 ⊢ (𝐾 ∪ {1, 𝑀}) = ((𝐾 ∪ {𝑀}) ∪ {1})
37	24, 31, 36	3eqtr4i 2762	. . 3 ⊢ ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = (𝐾 ∪ {1, 𝑀})
38		subfacp1lem1.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))
39	38	snssd 4773	. . . . . . 7 ⊢ (𝜑 → {𝑀} ⊆ (2...(𝑁 + 1)))
40		ssequn2 4152	. . . . . . 7 ⊢ ({𝑀} ⊆ (2...(𝑁 + 1)) ↔ ((2...(𝑁 + 1)) ∪ {𝑀}) = (2...(𝑁 + 1)))
41	39, 40	sylib 218	. . . . . 6 ⊢ (𝜑 → ((2...(𝑁 + 1)) ∪ {𝑀}) = (2...(𝑁 + 1)))
42		df-2 12249	. . . . . . 7 ⊢ 2 = (1 + 1)
43	42	oveq1i 7397	. . . . . 6 ⊢ (2...(𝑁 + 1)) = ((1 + 1)...(𝑁 + 1))
44	41, 43	eqtrdi 2780	. . . . 5 ⊢ (𝜑 → ((2...(𝑁 + 1)) ∪ {𝑀}) = ((1 + 1)...(𝑁 + 1)))
45	44	uneq2d 4131	. . . 4 ⊢ (𝜑 → ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = ((1...1) ∪ ((1 + 1)...(𝑁 + 1))))
46		subfacp1lem1.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47	46	peano2nnd 12203	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
48		nnuz 12836	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
49	47, 48	eleqtrdi 2838	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘1))
50		eluzfz1 13492	. . . . 5 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑁 + 1)))
51		fzsplit 13511	. . . . 5 ⊢ (1 ∈ (1...(𝑁 + 1)) → (1...(𝑁 + 1)) = ((1...1) ∪ ((1 + 1)...(𝑁 + 1))))
52	49, 50, 51	3syl 18	. . . 4 ⊢ (𝜑 → (1...(𝑁 + 1)) = ((1...1) ∪ ((1 + 1)...(𝑁 + 1))))
53	45, 52	eqtr4d 2767	. . 3 ⊢ (𝜑 → ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = (1...(𝑁 + 1)))
54	37, 53	eqtr3id 2778	. 2 ⊢ (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
55	42	oveq2i 7398	. . 3 ⊢ ((𝑁 + 1) − 2) = ((𝑁 + 1) − (1 + 1))
56		fzfi 13937	. . . . . . . . 9 ⊢ (2...(𝑁 + 1)) ∈ Fin
57		diffi 9139	. . . . . . . . 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 2824	. . . . . . 7 ⊢ 𝐾 ∈ Fin
60		prfi 9274	. . . . . . 7 ⊢ {1, 𝑀} ∈ Fin
61		hashun 14347	. . . . . . 7 ⊢ ((𝐾 ∈ Fin ∧ {1, 𝑀} ∈ Fin ∧ (𝐾 ∩ {1, 𝑀}) = ∅) → (♯‘(𝐾 ∪ {1, 𝑀})) = ((♯‘𝐾) + (♯‘{1, 𝑀})))
62	59, 60, 22, 61	mp3an 1463	. . . . . 6 ⊢ (♯‘(𝐾 ∪ {1, 𝑀})) = ((♯‘𝐾) + (♯‘{1, 𝑀}))
63	54	fveq2d 6862	. . . . . 6 ⊢ (𝜑 → (♯‘(𝐾 ∪ {1, 𝑀})) = (♯‘(1...(𝑁 + 1))))
64		neeq1 2987	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑥 ≠ 1 ↔ 𝑀 ≠ 1))
65	3, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2...(𝑁 + 1)) → 𝑥 ≠ 1)
66	64, 65	vtoclga 3543	. . . . . . . . . 10 ⊢ (𝑀 ∈ (2...(𝑁 + 1)) → 𝑀 ≠ 1)
67	38, 66	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 1)
68	67	necomd 2980	. . . . . . . 8 ⊢ (𝜑 → 1 ≠ 𝑀)
69		1ex 11170	. . . . . . . . 9 ⊢ 1 ∈ V
70		subfacp1lem1.x	. . . . . . . . 9 ⊢ 𝑀 ∈ V
71		hashprg 14360	. . . . . . . . 9 ⊢ ((1 ∈ V ∧ 𝑀 ∈ V) → (1 ≠ 𝑀 ↔ (♯‘{1, 𝑀}) = 2))
72	69, 70, 71	mp2an 692	. . . . . . . 8 ⊢ (1 ≠ 𝑀 ↔ (♯‘{1, 𝑀}) = 2)
73	68, 72	sylib 218	. . . . . . 7 ⊢ (𝜑 → (♯‘{1, 𝑀}) = 2)
74	73	oveq2d 7403	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐾) + (♯‘{1, 𝑀})) = ((♯‘𝐾) + 2))
75	62, 63, 74	3eqtr3a 2788	. . . . 5 ⊢ (𝜑 → (♯‘(1...(𝑁 + 1))) = ((♯‘𝐾) + 2))
76	47	nnnn0d 12503	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
77		hashfz1 14311	. . . . . 6 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
78	76, 77	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
79	75, 78	eqtr3d 2766	. . . 4 ⊢ (𝜑 → ((♯‘𝐾) + 2) = (𝑁 + 1))
80	47	nncnd 12202	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
81		2cnd 12264	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
82		hashcl 14321	. . . . . . . 8 ⊢ (𝐾 ∈ Fin → (♯‘𝐾) ∈ ℕ₀)
83	59, 82	ax-mp 5	. . . . . . 7 ⊢ (♯‘𝐾) ∈ ℕ₀
84	83	nn0cni 12454	. . . . . 6 ⊢ (♯‘𝐾) ∈ ℂ
85	84	a1i 11	. . . . 5 ⊢ (𝜑 → (♯‘𝐾) ∈ ℂ)
86	80, 81, 85	subadd2d 11552	. . . 4 ⊢ (𝜑 → (((𝑁 + 1) − 2) = (♯‘𝐾) ↔ ((♯‘𝐾) + 2) = (𝑁 + 1)))
87	79, 86	mpbird 257	. . 3 ⊢ (𝜑 → ((𝑁 + 1) − 2) = (♯‘𝐾))
88	46	nncnd 12202	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
89		1cnd 11169	. . . 4 ⊢ (𝜑 → 1 ∈ ℂ)
90	88, 89, 89	pnpcan2d 11571	. . 3 ⊢ (𝜑 → ((𝑁 + 1) − (1 + 1)) = (𝑁 − 1))
91	55, 87, 90	3eqtr3a 2788	. 2 ⊢ (𝜑 → (♯‘𝐾) = (𝑁 − 1))
92	23, 54, 91	3jca 1128	1 ⊢ (𝜑 → ((𝐾 ∩ {1, 𝑀}) = ∅ ∧ (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (♯‘𝐾) = (𝑁 − 1)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {cab 2707 ≠ wne 2925 ∀wral 3044 Vcvv 3447 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 {csn 4589 {cpr 4591 class class class wbr 5107 ↦ cmpt 5188 –1-1-onto→wf1o 6510 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 ℂcc 11066 1c1 11069 + caddc 11071 < clt 11208 ≤ cle 11209 − cmin 11405 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ...cfz 13468 ♯chash 14295

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 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 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-hash 14296

This theorem is referenced by: subfacp1lem2a 35167 subfacp1lem3 35169 subfacp1lem4 35170

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