Theorem iccpartres 48195

Description: The restriction of a partition is a partition. (Contributed by AV, 16-Jul-2020.)

Assertion

Ref	Expression
iccpartres	⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))

Proof of Theorem iccpartres

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2nn 12251	. . . 4 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
2		iccpart 48193	. . . 4 ⊢ ((𝑀 + 1) ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) ↔ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
3	1, 2	syl 18	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) ↔ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
4		simpl 487	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))))
5		nnz 12618	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
6		uzid 12883	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
7	5, 6	syl 18	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘𝑀))
8		peano2uz 12931	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀))
9	7, 8	syl 18	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ≥‘𝑀))
10		fzss2 13599	. . . . . . 7 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 1)))
11	9, 10	syl 18	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (0...𝑀) ⊆ (0...(𝑀 + 1)))
12		elmapssres 8862	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ (0...𝑀) ⊆ (0...(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ* ↑m (0...𝑀)))
13	4, 11, 12	syl2anr 608	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ* ↑m (0...𝑀)))
14		fzoss2 13723	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝑀) → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
15	9, 14	syl 18	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
16		ssralv 4005	. . . . . . . . 9 ⊢ ((0..^𝑀) ⊆ (0..^(𝑀 + 1)) → (∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
17	15, 16	syl 18	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
18	17	adantld 495	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
19	18	imp 411	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))
20		fzossfz 13714	. . . . . . . . . . . . . . 15 ⊢ (0..^𝑀) ⊆ (0...𝑀)
21	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) → (0..^𝑀) ⊆ (0...𝑀))
22	21	sselda 3936	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23		fvres 6900	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘𝑖) = (𝑃‘𝑖))
24	23	eqcomd 2768	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) = ((𝑃 ↾ (0...𝑀))‘𝑖))
25	22, 24	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) = ((𝑃 ↾ (0...𝑀))‘𝑖))
26		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
27		elfzouz 13699	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (ℤ≥‘0))
28	27	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (ℤ≥‘0))
29		fzofzp1b 13801	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (ℤ≥‘0) → (𝑖 ∈ (0..^𝑀) ↔ (𝑖 + 1) ∈ (0...𝑀)))
30	28, 29	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 ∈ (0..^𝑀) ↔ (𝑖 + 1) ∈ (0...𝑀)))
31	26, 30	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
32		fvres 6900	. . . . . . . . . . . . . 14 ⊢ ((𝑖 + 1) ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
33	31, 32	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
34	33	eqcomd 2768	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) = ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
35	25, 34	breq12d 5121	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) ↔ ((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
36	35	biimpd 232	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
37	36	ralimdva 3176	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
38	37	ex 417	. . . . . . . 8 ⊢ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) → (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
39	38	adantr 485	. . . . . . 7 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
40	39	impcom 412	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
41	19, 40	mpd 16	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
42		iccpart 48193	. . . . . 6 ⊢ (𝑀 ∈ ℕ → ((𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀) ↔ ((𝑃 ↾ (0...𝑀)) ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
43	42	adantr 485	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → ((𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀) ↔ ((𝑃 ↾ (0...𝑀)) ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
44	13, 41, 43	mpbir2and 725	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
45	44	ex 417	. . 3 ⊢ (𝑀 ∈ ℕ → ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
46	3, 45	sylbid 243	. 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
47	46	imp 411	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078   ⊆ wss 3904   class class class wbr 5108   ↾ cres 5662  ‘cfv 6536  (class class class)co 7412   ↑_m cmap 8822  0cc0 11106  1c1 11107   + caddc 11109  ℝ^*cxr 11248   < clt 11249  ℕcn 12239  ℤcz 12597  ℤ_≥cuz 12868  ...cfz 13541  ..^cfzo 13689  RePartciccp 48190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-n0 12511  df-z 12598  df-uz 12869  df-fz 13542  df-fzo 13690  df-iccp 48191

This theorem is used by:  iccelpart  48210