Theorem smndex1n0mnd 18947

Description: The identity of the monoid 𝑀 of endofunctions on set ℕ₀ is not contained in the base set of the constructed monoid 𝑆. (Contributed by AV, 17-Feb-2024.)

Hypotheses

Ref	Expression
smndex1ibas.m	⊢ 𝑀 = (EndoFMnd‘ℕ0)
smndex1ibas.n	⊢ 𝑁 ∈ ℕ
smndex1ibas.i	⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))
smndex1ibas.g	⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))
smndex1mgm.b	⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})
smndex1mgm.s	⊢ 𝑆 = (𝑀 ↾s 𝐵)

Assertion

Ref	Expression
smndex1n0mnd	⊢ (0g‘𝑀) ∉ 𝐵

Distinct variable groups:   𝑥,𝑁,𝑛   𝑥,𝑀   𝑛,𝐺   𝑛,𝑀   𝑥,𝐺   𝑛,𝐼,𝑥   𝑥,𝑆

Allowed substitution hints:   𝐵(𝑥,𝑛)   𝑆(𝑛)

Proof of Theorem smndex1n0mnd

Step	Hyp	Ref	Expression
1		smndex1ibas.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
2		nnnn0 12560	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
3		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑁 → (( I ↾ ℕ0)‘𝑥) = (( I ↾ ℕ0)‘𝑁))
4	1, 2	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝑁 ∈ ℕ0
5		fvresi 7207	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → (( I ↾ ℕ0)‘𝑁) = 𝑁)
6	4, 5	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( I ↾ ℕ0)‘𝑁) = 𝑁
7	3, 6	eqtrdi 2796	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → (( I ↾ ℕ0)‘𝑥) = 𝑁)
8		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → (𝐼‘𝑥) = (𝐼‘𝑁))
9	7, 8	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → ((( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥) ↔ 𝑁 = (𝐼‘𝑁)))
10	9	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (¬ (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥) ↔ ¬ 𝑁 = (𝐼‘𝑁)))
11	10	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 = 𝑁) → (¬ (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥) ↔ ¬ 𝑁 = (𝐼‘𝑁)))
12		nnne0 12327	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
13	12	neneqd 2951	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0)
14		smndex1ibas.i	. . . . . . . . . . 11 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))
15		oveq1 7455	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑁 → (𝑥 mod 𝑁) = (𝑁 mod 𝑁))
16		nnrp 13068	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+)
17		modid0 13948	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ+ → (𝑁 mod 𝑁) = 0)
18	16, 17	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝑁 mod 𝑁) = 0)
19	15, 18	sylan9eqr 2802	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 = 𝑁) → (𝑥 mod 𝑁) = 0)
20		c0ex 11284	. . . . . . . . . . . 12 ⊢ 0 ∈ V
21	20	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 0 ∈ V)
22	14, 19, 2, 21	fvmptd2 7037	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝐼‘𝑁) = 0)
23	22	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 = (𝐼‘𝑁) ↔ 𝑁 = 0))
24	13, 23	mtbird 325	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = (𝐼‘𝑁))
25	2, 11, 24	rspcedvd 3637	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ ℕ0 ¬ (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥))
26	1, 25	ax-mp 5	. . . . . 6 ⊢ ∃𝑥 ∈ ℕ0 ¬ (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥)
27		rexnal 3106	. . . . . 6 ⊢ (∃𝑥 ∈ ℕ0 ¬ (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥) ↔ ¬ ∀𝑥 ∈ ℕ0 (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥))
28	26, 27	mpbi 230	. . . . 5 ⊢ ¬ ∀𝑥 ∈ ℕ0 (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥)
29		fnresi 6709	. . . . . 6 ⊢ ( I ↾ ℕ0) Fn ℕ0
30		ovex 7481	. . . . . . 7 ⊢ (𝑥 mod 𝑁) ∈ V
31	30, 14	fnmpti 6723	. . . . . 6 ⊢ 𝐼 Fn ℕ0
32		eqfnfv 7064	. . . . . 6 ⊢ ((( I ↾ ℕ0) Fn ℕ0 ∧ 𝐼 Fn ℕ0) → (( I ↾ ℕ0) = 𝐼 ↔ ∀𝑥 ∈ ℕ0 (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥)))
33	29, 31, 32	mp2an 691	. . . . 5 ⊢ (( I ↾ ℕ0) = 𝐼 ↔ ∀𝑥 ∈ ℕ0 (( I ↾ ℕ0)‘𝑥) = (𝐼‘𝑥))
34	28, 33	mtbir 323	. . . 4 ⊢ ¬ ( I ↾ ℕ0) = 𝐼
35	4	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑁 ∈ ℕ0)
36		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑁 → ((𝐺‘𝑛)‘𝑥) = ((𝐺‘𝑛)‘𝑁))
37	7, 36	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑥 = 𝑁 → ((( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥) ↔ 𝑁 = ((𝐺‘𝑛)‘𝑁)))
38	37	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → (¬ (( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥) ↔ ¬ 𝑁 = ((𝐺‘𝑛)‘𝑁)))
39	38	adantl 481	. . . . . . . 8 ⊢ ((𝑛 ∈ (0..^𝑁) ∧ 𝑥 = 𝑁) → (¬ (( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥) ↔ ¬ 𝑁 = ((𝐺‘𝑛)‘𝑁)))
40		fzonel 13730	. . . . . . . . . . 11 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
41		eleq1 2832	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (0..^𝑁) ↔ 𝑁 ∈ (0..^𝑁)))
42	41	eqcoms 2748	. . . . . . . . . . 11 ⊢ (𝑁 = 𝑛 → (𝑛 ∈ (0..^𝑁) ↔ 𝑁 ∈ (0..^𝑁)))
43	40, 42	mtbiri 327	. . . . . . . . . 10 ⊢ (𝑁 = 𝑛 → ¬ 𝑛 ∈ (0..^𝑁))
44	43	con2i 139	. . . . . . . . 9 ⊢ (𝑛 ∈ (0..^𝑁) → ¬ 𝑁 = 𝑛)
45		nn0ex 12559	. . . . . . . . . . . . 13 ⊢ ℕ0 ∈ V
46	45	mptex 7260	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0 ↦ 𝑛) ∈ V
47		smndex1ibas.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))
48	47	fvmpt2 7040	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (0..^𝑁) ∧ (𝑥 ∈ ℕ0 ↦ 𝑛) ∈ V) → (𝐺‘𝑛) = (𝑥 ∈ ℕ0 ↦ 𝑛))
49	46, 48	mpan2 690	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0..^𝑁) → (𝐺‘𝑛) = (𝑥 ∈ ℕ0 ↦ 𝑛))
50		eqidd 2741	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (0..^𝑁) ∧ 𝑥 = 𝑁) → 𝑛 = 𝑛)
51		id 22	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ (0..^𝑁))
52	49, 50, 35, 51	fvmptd 7036	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0..^𝑁) → ((𝐺‘𝑛)‘𝑁) = 𝑛)
53	52	eqeq2d 2751	. . . . . . . . 9 ⊢ (𝑛 ∈ (0..^𝑁) → (𝑁 = ((𝐺‘𝑛)‘𝑁) ↔ 𝑁 = 𝑛))
54	44, 53	mtbird 325	. . . . . . . 8 ⊢ (𝑛 ∈ (0..^𝑁) → ¬ 𝑁 = ((𝐺‘𝑛)‘𝑁))
55	35, 39, 54	rspcedvd 3637	. . . . . . 7 ⊢ (𝑛 ∈ (0..^𝑁) → ∃𝑥 ∈ ℕ0 ¬ (( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥))
56		rexnal 3106	. . . . . . 7 ⊢ (∃𝑥 ∈ ℕ0 ¬ (( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥) ↔ ¬ ∀𝑥 ∈ ℕ0 (( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥))
57	55, 56	sylib 218	. . . . . 6 ⊢ (𝑛 ∈ (0..^𝑁) → ¬ ∀𝑥 ∈ ℕ0 (( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥))
58		vex 3492	. . . . . . . . 9 ⊢ 𝑛 ∈ V
59		eqid 2740	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 ↦ 𝑛) = (𝑥 ∈ ℕ0 ↦ 𝑛)
60	58, 59	fnmpti 6723	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0 ↦ 𝑛) Fn ℕ0
61	49	fneq1d 6672	. . . . . . . 8 ⊢ (𝑛 ∈ (0..^𝑁) → ((𝐺‘𝑛) Fn ℕ0 ↔ (𝑥 ∈ ℕ0 ↦ 𝑛) Fn ℕ0))
62	60, 61	mpbiri 258	. . . . . . 7 ⊢ (𝑛 ∈ (0..^𝑁) → (𝐺‘𝑛) Fn ℕ0)
63		eqfnfv 7064	. . . . . . 7 ⊢ ((( I ↾ ℕ0) Fn ℕ0 ∧ (𝐺‘𝑛) Fn ℕ0) → (( I ↾ ℕ0) = (𝐺‘𝑛) ↔ ∀𝑥 ∈ ℕ0 (( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥)))
64	29, 62, 63	sylancr 586	. . . . . 6 ⊢ (𝑛 ∈ (0..^𝑁) → (( I ↾ ℕ0) = (𝐺‘𝑛) ↔ ∀𝑥 ∈ ℕ0 (( I ↾ ℕ0)‘𝑥) = ((𝐺‘𝑛)‘𝑥)))
65	57, 64	mtbird 325	. . . . 5 ⊢ (𝑛 ∈ (0..^𝑁) → ¬ ( I ↾ ℕ0) = (𝐺‘𝑛))
66	65	nrex 3080	. . . 4 ⊢ ¬ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛)
67	34, 66	pm3.2ni 879	. . 3 ⊢ ¬ (( I ↾ ℕ0) = 𝐼 ∨ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛))
68		smndex1ibas.m	. . . . . . . 8 ⊢ 𝑀 = (EndoFMnd‘ℕ0)
69	68	efmndid 18923	. . . . . . 7 ⊢ (ℕ0 ∈ V → ( I ↾ ℕ0) = (0g‘𝑀))
70	45, 69	ax-mp 5	. . . . . 6 ⊢ ( I ↾ ℕ0) = (0g‘𝑀)
71	70	eqcomi 2749	. . . . 5 ⊢ (0g‘𝑀) = ( I ↾ ℕ0)
72		smndex1mgm.b	. . . . 5 ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})
73	71, 72	eleq12i 2837	. . . 4 ⊢ ((0g‘𝑀) ∈ 𝐵 ↔ ( I ↾ ℕ0) ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
74		elun 4176	. . . . 5 ⊢ (( I ↾ ℕ0) ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (( I ↾ ℕ0) ∈ {𝐼} ∨ ( I ↾ ℕ0) ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
75		resiexg 7952	. . . . . . . 8 ⊢ (ℕ0 ∈ V → ( I ↾ ℕ0) ∈ V)
76	45, 75	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ ℕ0) ∈ V
77	76	elsn 4663	. . . . . 6 ⊢ (( I ↾ ℕ0) ∈ {𝐼} ↔ ( I ↾ ℕ0) = 𝐼)
78		eliun 5019	. . . . . . 7 ⊢ (( I ↾ ℕ0) ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ↔ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) ∈ {(𝐺‘𝑛)})
79	76	elsn 4663	. . . . . . . 8 ⊢ (( I ↾ ℕ0) ∈ {(𝐺‘𝑛)} ↔ ( I ↾ ℕ0) = (𝐺‘𝑛))
80	79	rexbii 3100	. . . . . . 7 ⊢ (∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) ∈ {(𝐺‘𝑛)} ↔ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛))
81	78, 80	bitri 275	. . . . . 6 ⊢ (( I ↾ ℕ0) ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} ↔ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛))
82	77, 81	orbi12i 913	. . . . 5 ⊢ ((( I ↾ ℕ0) ∈ {𝐼} ∨ ( I ↾ ℕ0) ∈ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (( I ↾ ℕ0) = 𝐼 ∨ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛)))
83	74, 82	bitri 275	. . . 4 ⊢ (( I ↾ ℕ0) ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (( I ↾ ℕ0) = 𝐼 ∨ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛)))
84	73, 83	bitri 275	. . 3 ⊢ ((0g‘𝑀) ∈ 𝐵 ↔ (( I ↾ ℕ0) = 𝐼 ∨ ∃𝑛 ∈ (0..^𝑁)( I ↾ ℕ0) = (𝐺‘𝑛)))
85	67, 84	mtbir 323	. 2 ⊢ ¬ (0g‘𝑀) ∈ 𝐵
86	85	nelir 3055	1 ⊢ (0g‘𝑀) ∉ 𝐵

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ∉ wnel 3052  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∪ cun 3974  {csn 4648  ∪ ciun 5015   ↦ cmpt 5249   I cid 5592   ↾ cres 5702   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  0cc0 11184  ℕcn 12293  ℕ₀cn0 12553  ℝ⁺crp 13057  ..^cfzo 13711   mod cmo 13920   ↾_s cress 17287  0_gc0g 17499  EndoFMndcefmnd 18903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-fl 13843  df-mod 13921  df-struct 17194  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-tset 17330  df-0g 17501  df-efmnd 18904

This theorem is referenced by:  nsmndex1  18948