Theorem iccpartres 42242

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 11364	. . . 4 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
2		iccpart 42240	. . . 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 476	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))))
5		nnz 11727	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
6		uzid 11983	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘𝑀))
8		peano2uz 12023	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ≥‘𝑀))
10		fzss2 12674	. . . . . . 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 8147	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ (0...𝑀) ⊆ (0...(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ* ↑𝑚 (0...𝑀)))
13	4, 11, 12	syl2anr 592	. . . . 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 12791	. . . . . . . . . 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 3891	. . . . . . . . 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 486	. . . . . . 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 397	. . . . . 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 12783	. . . . . . . . . . . . . . 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 3827	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23		fvres 6452	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘𝑖) = (𝑃‘𝑖))
24	23	eqcomd 2831	. . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
27		elfzouz 12769	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (ℤ≥‘0))
28	27	adantl 475	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (ℤ≥‘0))
29		fzofzp1b 12861	. . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
32		fvres 6452	. . . . . . . . . . . . . 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 2831	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℝ* ↑𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) = ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
35	25, 34	breq12d 4886	. . . . . . . . . . 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 221	. . . . . . . . . 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 3171	. . . . . . . . 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 403	. . . . . . . 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 474	. . . . . . 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 398	. . . . . 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 42240	. . . . . 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 474	. . . . 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 706	. . . 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 403	. . 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 232	. 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
47	46	imp 397	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117   ⊆ wss 3798   class class class wbr 4873   ↾ cres 5344  ‘cfv 6123  (class class class)co 6905   ↑_𝑚 cmap 8122  0cc0 10252  1c1 10253   + caddc 10255  ℝ^*cxr 10390   < clt 10391  ℕcn 11350  ℤcz 11704  ℤ_≥cuz 11968  ...cfz 12619  ..^cfzo 12760  RePartciccp 42237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-er 8009  df-map 8124  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-n0 11619  df-z 11705  df-uz 11969  df-fz 12620  df-fzo 12761  df-iccp 42238

This theorem is referenced by:  iccelpart  42257