subfacp1lem1 - Mathbox for Mario Carneiro

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Theorem subfacp1lem1 32421

Description: Lemma for subfacp1 32428. 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 4398	. . . 4 ⊢ ((𝐾 ∩ {1, 𝑀}) = ∅ ↔ ∀𝑥 ∈ 𝐾 ¬ 𝑥 ∈ {1, 𝑀})
2		eldifi 4102	. . . . . . . . 9 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → 𝑥 ∈ (2...(𝑁 + 1)))
3		elfzle1 12904	. . . . . . . . 9 ⊢ (𝑥 ∈ (2...(𝑁 + 1)) → 2 ≤ 𝑥)
4		1lt2 11802	. . . . . . . . . . . 12 ⊢ 1 < 2
5		1re 10635	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
6		2re 11705	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
7	5, 6	ltnlei 10755	. . . . . . . . . . . 12 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
8	4, 7	mpbi 232	. . . . . . . . . . 11 ⊢ ¬ 2 ≤ 1
9		breq2 5062	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (2 ≤ 𝑥 ↔ 2 ≤ 1))
10	8, 9	mtbiri 329	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ¬ 2 ≤ 𝑥)
11	10	necon2ai 3045	. . . . . . . . 9 ⊢ (2 ≤ 𝑥 → 𝑥 ≠ 1)
12	2, 3, 11	3syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → 𝑥 ≠ 1)
13		eldifsni 4715	. . . . . . . 8 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → 𝑥 ≠ 𝑀)
14	12, 13	jca 514	. . . . . . 7 ⊢ (𝑥 ∈ ((2...(𝑁 + 1)) ∖ {𝑀}) → (𝑥 ≠ 1 ∧ 𝑥 ≠ 𝑀))
15		subfacp1lem1.k	. . . . . . 7 ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀})
16	14, 15	eleq2s 2931	. . . . . 6 ⊢ (𝑥 ∈ 𝐾 → (𝑥 ≠ 1 ∧ 𝑥 ≠ 𝑀))
17		neanior 3109	. . . . . 6 ⊢ ((𝑥 ≠ 1 ∧ 𝑥 ≠ 𝑀) ↔ ¬ (𝑥 = 1 ∨ 𝑥 = 𝑀))
18	16, 17	sylib 220	. . . . 5 ⊢ (𝑥 ∈ 𝐾 → ¬ (𝑥 = 1 ∨ 𝑥 = 𝑀))
19		vex 3497	. . . . . 6 ⊢ 𝑥 ∈ V
20	19	elpr 4583	. . . . 5 ⊢ (𝑥 ∈ {1, 𝑀} ↔ (𝑥 = 1 ∨ 𝑥 = 𝑀))
21	18, 20	sylnibr 331	. . . 4 ⊢ (𝑥 ∈ 𝐾 → ¬ 𝑥 ∈ {1, 𝑀})
22	1, 21	mprgbir 3153	. . 3 ⊢ (𝐾 ∩ {1, 𝑀}) = ∅
23	22	a1i 11	. 2 ⊢ (𝜑 → (𝐾 ∩ {1, 𝑀}) = ∅)
24		uncom 4128	. . . 4 ⊢ ({1} ∪ (𝐾 ∪ {𝑀})) = ((𝐾 ∪ {𝑀}) ∪ {1})
25		1z 12006	. . . . . 6 ⊢ 1 ∈ ℤ
26		fzsn 12943	. . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1})
27	25, 26	ax-mp 5	. . . . 5 ⊢ (1...1) = {1}
28	15	uneq1i 4134	. . . . . 6 ⊢ (𝐾 ∪ {𝑀}) = (((2...(𝑁 + 1)) ∖ {𝑀}) ∪ {𝑀})
29		undif1 4423	. . . . . 6 ⊢ (((2...(𝑁 + 1)) ∖ {𝑀}) ∪ {𝑀}) = ((2...(𝑁 + 1)) ∪ {𝑀})
30	28, 29	eqtr2i 2845	. . . . 5 ⊢ ((2...(𝑁 + 1)) ∪ {𝑀}) = (𝐾 ∪ {𝑀})
31	27, 30	uneq12i 4136	. . . 4 ⊢ ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = ({1} ∪ (𝐾 ∪ {𝑀}))
32		df-pr 4563	. . . . . . 7 ⊢ {1, 𝑀} = ({1} ∪ {𝑀})
33	32	equncomi 4130	. . . . . 6 ⊢ {1, 𝑀} = ({𝑀} ∪ {1})
34	33	uneq2i 4135	. . . . 5 ⊢ (𝐾 ∪ {1, 𝑀}) = (𝐾 ∪ ({𝑀} ∪ {1}))
35		unass 4141	. . . . 5 ⊢ ((𝐾 ∪ {𝑀}) ∪ {1}) = (𝐾 ∪ ({𝑀} ∪ {1}))
36	34, 35	eqtr4i 2847	. . . 4 ⊢ (𝐾 ∪ {1, 𝑀}) = ((𝐾 ∪ {𝑀}) ∪ {1})
37	24, 31, 36	3eqtr4i 2854	. . 3 ⊢ ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = (𝐾 ∪ {1, 𝑀})
38		subfacp1lem1.m	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1)))
39	38	snssd 4735	. . . . . . 7 ⊢ (𝜑 → {𝑀} ⊆ (2...(𝑁 + 1)))
40		ssequn2 4158	. . . . . . 7 ⊢ ({𝑀} ⊆ (2...(𝑁 + 1)) ↔ ((2...(𝑁 + 1)) ∪ {𝑀}) = (2...(𝑁 + 1)))
41	39, 40	sylib 220	. . . . . 6 ⊢ (𝜑 → ((2...(𝑁 + 1)) ∪ {𝑀}) = (2...(𝑁 + 1)))
42		df-2 11694	. . . . . . 7 ⊢ 2 = (1 + 1)
43	42	oveq1i 7160	. . . . . 6 ⊢ (2...(𝑁 + 1)) = ((1 + 1)...(𝑁 + 1))
44	41, 43	syl6eq 2872	. . . . 5 ⊢ (𝜑 → ((2...(𝑁 + 1)) ∪ {𝑀}) = ((1 + 1)...(𝑁 + 1)))
45	44	uneq2d 4138	. . . 4 ⊢ (𝜑 → ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = ((1...1) ∪ ((1 + 1)...(𝑁 + 1))))
46		subfacp1lem1.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47	46	peano2nnd 11649	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
48		nnuz 12275	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
49	47, 48	eleqtrdi 2923	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘1))
50		eluzfz1 12908	. . . . 5 ⊢ ((𝑁 + 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑁 + 1)))
51		fzsplit 12927	. . . . 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 2859	. . 3 ⊢ (𝜑 → ((1...1) ∪ ((2...(𝑁 + 1)) ∪ {𝑀})) = (1...(𝑁 + 1)))
54	37, 53	syl5eqr 2870	. 2 ⊢ (𝜑 → (𝐾 ∪ {1, 𝑀}) = (1...(𝑁 + 1)))
55	42	oveq2i 7161	. . 3 ⊢ ((𝑁 + 1) − 2) = ((𝑁 + 1) − (1 + 1))
56		fzfi 13334	. . . . . . . . 9 ⊢ (2...(𝑁 + 1)) ∈ Fin
57		diffi 8744	. . . . . . . . 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 2909	. . . . . . 7 ⊢ 𝐾 ∈ Fin
60		prfi 8787	. . . . . . 7 ⊢ {1, 𝑀} ∈ Fin
61		hashun 13737	. . . . . . 7 ⊢ ((𝐾 ∈ Fin ∧ {1, 𝑀} ∈ Fin ∧ (𝐾 ∩ {1, 𝑀}) = ∅) → (♯‘(𝐾 ∪ {1, 𝑀})) = ((♯‘𝐾) + (♯‘{1, 𝑀})))
62	59, 60, 22, 61	mp3an 1457	. . . . . 6 ⊢ (♯‘(𝐾 ∪ {1, 𝑀})) = ((♯‘𝐾) + (♯‘{1, 𝑀}))
63	54	fveq2d 6668	. . . . . 6 ⊢ (𝜑 → (♯‘(𝐾 ∪ {1, 𝑀})) = (♯‘(1...(𝑁 + 1))))
64		neeq1 3078	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑀 → (𝑥 ≠ 1 ↔ 𝑀 ≠ 1))
65	3, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2...(𝑁 + 1)) → 𝑥 ≠ 1)
66	64, 65	vtoclga 3573	. . . . . . . . . 10 ⊢ (𝑀 ∈ (2...(𝑁 + 1)) → 𝑀 ≠ 1)
67	38, 66	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ≠ 1)
68	67	necomd 3071	. . . . . . . 8 ⊢ (𝜑 → 1 ≠ 𝑀)
69		1ex 10631	. . . . . . . . 9 ⊢ 1 ∈ V
70		subfacp1lem1.x	. . . . . . . . 9 ⊢ 𝑀 ∈ V
71		hashprg 13750	. . . . . . . . 9 ⊢ ((1 ∈ V ∧ 𝑀 ∈ V) → (1 ≠ 𝑀 ↔ (♯‘{1, 𝑀}) = 2))
72	69, 70, 71	mp2an 690	. . . . . . . 8 ⊢ (1 ≠ 𝑀 ↔ (♯‘{1, 𝑀}) = 2)
73	68, 72	sylib 220	. . . . . . 7 ⊢ (𝜑 → (♯‘{1, 𝑀}) = 2)
74	73	oveq2d 7166	. . . . . 6 ⊢ (𝜑 → ((♯‘𝐾) + (♯‘{1, 𝑀})) = ((♯‘𝐾) + 2))
75	62, 63, 74	3eqtr3a 2880	. . . . 5 ⊢ (𝜑 → (♯‘(1...(𝑁 + 1))) = ((♯‘𝐾) + 2))
76	47	nnnn0d 11949	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
77		hashfz1 13700	. . . . . 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 2858	. . . 4 ⊢ (𝜑 → ((♯‘𝐾) + 2) = (𝑁 + 1))
80	47	nncnd 11648	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
81		2cnd 11709	. . . . 5 ⊢ (𝜑 → 2 ∈ ℂ)
82		hashcl 13711	. . . . . . . 8 ⊢ (𝐾 ∈ Fin → (♯‘𝐾) ∈ ℕ₀)
83	59, 82	ax-mp 5	. . . . . . 7 ⊢ (♯‘𝐾) ∈ ℕ₀
84	83	nn0cni 11903	. . . . . 6 ⊢ (♯‘𝐾) ∈ ℂ
85	84	a1i 11	. . . . 5 ⊢ (𝜑 → (♯‘𝐾) ∈ ℂ)
86	80, 81, 85	subadd2d 11010	. . . 4 ⊢ (𝜑 → (((𝑁 + 1) − 2) = (♯‘𝐾) ↔ ((♯‘𝐾) + 2) = (𝑁 + 1)))
87	79, 86	mpbird 259	. . 3 ⊢ (𝜑 → ((𝑁 + 1) − 2) = (♯‘𝐾))
88	46	nncnd 11648	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℂ)
89		1cnd 10630	. . . 4 ⊢ (𝜑 → 1 ∈ ℂ)
90	88, 89, 89	pnpcan2d 11029	. . 3 ⊢ (𝜑 → ((𝑁 + 1) − (1 + 1)) = (𝑁 − 1))
91	55, 87, 90	3eqtr3a 2880	. 2 ⊢ (𝜑 → (♯‘𝐾) = (𝑁 − 1))
92	23, 54, 91	3jca 1124	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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4560 {cpr 4562 class class class wbr 5058 ↦ cmpt 5138 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 ℂcc 10529 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 − cmin 10864 ℕcn 11632 2c2 11686 ℕ₀cn0 11891 ℤcz 11975 ℤ_≥cuz 12237 ...cfz 12886 ♯chash 13684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685

This theorem is referenced by: subfacp1lem2a 32422 subfacp1lem3 32424 subfacp1lem4 32425

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