Theorem uzrdgsuc 4659

Description: Successor value of a recursive definition generator on an upper partition of ℤ. See comment in uzrdgval 4657.

Hypotheses

Ref	Expression
om2uz.1	⊢ C ∈ ℤ
om2uz.2	⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω)

Assertion

Ref	Expression
uzrdgsuc	⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘(B + 1)) = (F ‘((rec(F, A) ∘ ◡G) ‘B)))

Distinct variable group(s):   x,y,z   z,G   z,A   z,B   x,C,y,z

Proof of Theorem uzrdgsuc

Step	Hyp	Ref	Expression
1		om2uz.1	. . . . 5 ⊢ C ∈ ℤ
2		peano2uz 4602	. . . . 5 ⊢ ((C ∈ ℤ ∧ B ∈ {z ∈ ℤ∣C ≤ z}) → (B + 1) ∈ {z ∈ ℤ∣C ≤ z})
3	1, 2	mpan 518	. . . 4 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (B + 1) ∈ {z ∈ ℤ∣C ≤ z})
4		om2uz.2	. . . . 5 ⊢ G = (rec({⟨x, y⟩∣y = (x + 1)}, C) ↾ ω)
5	1, 4	uzrdgval 4657	. . . 4 ⊢ ((B + 1) ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘(B + 1)) = (rec(F, A) ‘(◡G ‘(B + 1))))
6	3, 5	syl 12	. . 3 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘(B + 1)) = (rec(F, A) ‘(◡G ‘(B + 1))))
7	1, 4	om2uzf1o 4656	. . . . . . . . . 10 ⊢ G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z}
8		f1ocnv 2811	. . . . . . . . . 10 ⊢ (G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z} → ◡G:{z ∈ ℤ∣C ≤ z}–1-1-onto→ω)
9	7, 8	ax-mp 6	. . . . . . . . 9 ⊢ ◡G:{z ∈ ℤ∣C ≤ z}–1-1-onto→ω
10		f1of 2800	. . . . . . . . 9 ⊢ (◡G:{z ∈ ℤ∣C ≤ z}–1-1-onto→ω → ◡G:{z ∈ ℤ∣C ≤ z}–→ω)
11	9, 10	ax-mp 6	. . . . . . . 8 ⊢ ◡G:{z ∈ ℤ∣C ≤ z}–→ω
12		ffvrn 2890	. . . . . . . 8 ⊢ ((◡G:{z ∈ ℤ∣C ≤ z}–→ω ∧ B ∈ {z ∈ ℤ∣C ≤ z}) → (◡G ‘B) ∈ ω)
13	11, 12	mpan 518	. . . . . . 7 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (◡G ‘B) ∈ ω)
14	1, 4	om2uzsuc 4652	. . . . . . 7 ⊢ ((◡G ‘B) ∈ ω → (G ‘suc (◡G ‘B)) = ((G ‘(◡G ‘B)) + 1))
15	13, 14	syl 12	. . . . . 6 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (G ‘suc (◡G ‘B)) = ((G ‘(◡G ‘B)) + 1))
16		f1ocnvfv2 2920	. . . . . . . 8 ⊢ ((G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z} ∧ B ∈ {z ∈ ℤ∣C ≤ z}) → (G ‘(◡G ‘B)) = B)
17	7, 16	mpan 518	. . . . . . 7 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (G ‘(◡G ‘B)) = B)
18	17	opreq1d 3012	. . . . . 6 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((G ‘(◡G ‘B)) + 1) = (B + 1))
19	15, 18	eqtrd 1128	. . . . 5 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (G ‘suc (◡G ‘B)) = (B + 1))
20		peano2 2391	. . . . . 6 ⊢ ((◡G ‘B) ∈ ω → suc (◡G ‘B) ∈ ω)
21		f1ocnvfv 2921	. . . . . . 7 ⊢ ((G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z} ∧ suc (◡G ‘B) ∈ ω) → ((G ‘suc (◡G ‘B)) = (B + 1) → (◡G ‘(B + 1)) = suc (◡G ‘B)))
22	7, 21	mpan 518	. . . . . 6 ⊢ (suc (◡G ‘B) ∈ ω → ((G ‘suc (◡G ‘B)) = (B + 1) → (◡G ‘(B + 1)) = suc (◡G ‘B)))
23	13, 20, 22	3syl 21	. . . . 5 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((G ‘suc (◡G ‘B)) = (B + 1) → (◡G ‘(B + 1)) = suc (◡G ‘B)))
24	19, 23	mpd 46	. . . 4 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (◡G ‘(B + 1)) = suc (◡G ‘B))
25	24	fveq2d 2836	. . 3 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (rec(F, A) ‘(◡G ‘(B + 1))) = (rec(F, A) ‘suc (◡G ‘B)))
26		nnont 2379	. . . 4 ⊢ ((◡G ‘B) ∈ ω → (◡G ‘B) ∈ On)
27		rdgsuct 2983	. . . 4 ⊢ ((◡G ‘B) ∈ On → (rec(F, A) ‘suc (◡G ‘B)) = (F ‘(rec(F, A) ‘(◡G ‘B))))
28	13, 26, 27	3syl 21	. . 3 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (rec(F, A) ‘suc (◡G ‘B)) = (F ‘(rec(F, A) ‘(◡G ‘B))))
29	6, 25, 28	3eqtrd 1132	. 2 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘(B + 1)) = (F ‘(rec(F, A) ‘(◡G ‘B))))
30	1, 4	uzrdgval 4657	. . 3 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘B) = (rec(F, A) ‘(◡G ‘B)))
31	30	fveq2d 2836	. 2 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (F ‘((rec(F, A) ∘ ◡G) ‘B)) = (F ‘(rec(F, A) ‘(◡G ‘B))))
32	29, 31	eqtr4d 1131	1 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘(B + 1)) = (F ‘((rec(F, A) ∘ ◡G) ‘B)))

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  {crab 1204   class class class wbr 2054  {copab 2055  Oncon0 2199  suc csuc 2201  ωcom 2372  ^◡ccnv 2409   ↾ cres 2412   ∘ ccom 2414  –→wf 2418  –1-1-onto→wf1o 2421   ‘cfv 2422  reccrdg 2969  (class class class)co 3001  1c1 4029   + caddc 4031   ≤ cle 4092  ℤcz 4095

This theorem is referenced by:  seqlem1 4662  seqsuclem 4669

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-le 4277  df-n 4423  df-n0 4535  df-z 4564

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