Theorem iccpartres 41354

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 11032	. . . 4 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
2		iccpart 41352	. . . 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 17	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) ↔ (𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
4		simpl 473	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))))
5		nnz 11399	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
6		uzid 11702	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘𝑀))
8		peano2uz 11741	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ≥‘𝑀))
10		fzss2 12381	. . . . . . 7 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 1)))
11	9, 10	syl 17	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (0...𝑀) ⊆ (0...(𝑀 + 1)))
12		elmapssres 7882	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ (0...𝑀) ⊆ (0...(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ* ↑𝑚 (0...𝑀)))
13	4, 11, 12	syl2anr 495	. . . . 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 12496	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝑀) → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
15	9, 14	syl 17	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
16		ssralv 3666	. . . . . . . . 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 17	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
18	17	adantld 483	. . . . . . 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 445	. . . . . 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 12488	. . . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23		fvres 6207	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘𝑖) = (𝑃‘𝑖))
24	23	eqcomd 2628	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) = ((𝑃 ↾ (0...𝑀))‘𝑖))
25	22, 24	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) = ((𝑃 ↾ (0...𝑀))‘𝑖))
26		simpr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
27		elfzouz 12474	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (ℤ≥‘0))
28	27	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (ℤ≥‘0))
29		fzofzp1b 12566	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (ℤ≥‘0) → (𝑖 ∈ (0..^𝑀) ↔ (𝑖 + 1) ∈ (0...𝑀)))
30	28, 29	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 ∈ (0..^𝑀) ↔ (𝑖 + 1) ∈ (0...𝑀)))
31	26, 30	mpbid 222	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
32		fvres 6207	. . . . . . . . . . . . . 14 ⊢ ((𝑖 + 1) ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
34	33	eqcomd 2628	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) = ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
35	25, 34	breq12d 4666	. . . . . . . . . . 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 219	. . . . . . . . . 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 2962	. . . . . . . . 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 450	. . . . . . . 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 481	. . . . . . 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 446	. . . . . 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 15	. . . . 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 41352	. . . . . 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 481	. . . . 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 957	. . . 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 450	. . 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 230	. 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
47	46	imp 445	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574   class class class wbr 4653   ↾ cres 5116  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  0cc0 9936  1c1 9937   + caddc 9939  ℝ^*cxr 10073   < clt 10074  ℕcn 11020  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ..^cfzo 12465  RePartciccp 41349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-iccp 41350

This theorem is referenced by:  iccelpart  41369