Theorem uzrdgsuc 6305

Description: Successor value of a recursive definition generator on upper integers. See comment in uzrdgval 6303.

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 groups:   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		peano2uz2 6203	. . . . 5 ⊢ ((C ∈ ℤ ⋀ B ∈ {z ∈ ℤ∣C ≤ z}) → (B + 1) ∈ {z ∈ ℤ∣C ≤ z})
3	1, 2	mpan 697	. . . 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 6303	. . . 4 ⊢ ((B + 1) ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘(B + 1)) = (rec(F, A) ‘(◡G ‘(B + 1))))
6	3, 5	syl 10	. . 3 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘(B + 1)) = (rec(F, A) ‘(◡G ‘(B + 1))))
7	1, 4	om2uzf1o 6302	. . . . . . . . . 10 ⊢ G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z}
8		f1ocnv 3707	. . . . . . . . . 10 ⊢ (G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z} → ◡G:{z ∈ ℤ∣C ≤ z}–1-1-onto→ω)
9	7, 8	ax-mp 7	. . . . . . . . 9 ⊢ ◡G:{z ∈ ℤ∣C ≤ z}–1-1-onto→ω
10		f1of 3695	. . . . . . . . 9 ⊢ (◡G:{z ∈ ℤ∣C ≤ z}–1-1-onto→ω → ◡G:{z ∈ ℤ∣C ≤ z}–→ω)
11	9, 10	ax-mp 7	. . . . . . . 8 ⊢ ◡G:{z ∈ ℤ∣C ≤ z}–→ω
12	11	ffvelrni 3821	. . . . . . 7 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (◡G ‘B) ∈ ω)
13	1, 4	om2uzsuc 6297	. . . . . . 7 ⊢ ((◡G ‘B) ∈ ω → (G ‘suc (◡G ‘B)) = ((G ‘(◡G ‘B)) + 1))
14	12, 13	syl 10	. . . . . 6 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (G ‘suc (◡G ‘B)) = ((G ‘(◡G ‘B)) + 1))
15		f1ocnvfv2 3885	. . . . . . . 8 ⊢ ((G:ω–1-1-onto→{z ∈ ℤ∣C ≤ z} ⋀ B ∈ {z ∈ ℤ∣C ≤ z}) → (G ‘(◡G ‘B)) = B)
16	7, 15	mpan 697	. . . . . . 7 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (G ‘(◡G ‘B)) = B)
17	16	opreq1d 3981	. . . . . 6 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((G ‘(◡G ‘B)) + 1) = (B + 1))
18	14, 17	eqtrd 1510	. . . . 5 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (G ‘suc (◡G ‘B)) = (B + 1))
19		peano2 3156	. . . . . 6 ⊢ ((◡G ‘B) ∈ ω → suc (◡G ‘B) ∈ ω)
20		f1ocnvfv 3886	. . . . . . 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)))
21	7, 20	mpan 697	. . . . . 6 ⊢ (suc (◡G ‘B) ∈ ω → ((G ‘suc (◡G ‘B)) = (B + 1) → (◡G ‘(B + 1)) = suc (◡G ‘B)))
22	12, 19, 21	3syl 20	. . . . 5 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((G ‘suc (◡G ‘B)) = (B + 1) → (◡G ‘(B + 1)) = suc (◡G ‘B)))
23	18, 22	mpd 26	. . . 4 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (◡G ‘(B + 1)) = suc (◡G ‘B))
24	23	fveq2d 3734	. . 3 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (rec(F, A) ‘(◡G ‘(B + 1))) = (rec(F, A) ‘suc (◡G ‘B)))
25		nnont 3144	. . . 4 ⊢ ((◡G ‘B) ∈ ω → (◡G ‘B) ∈ On)
26		rdgsuct 3951	. . . 4 ⊢ ((◡G ‘B) ∈ On → (rec(F, A) ‘suc (◡G ‘B)) = (F ‘(rec(F, A) ‘(◡G ‘B))))
27	12, 25, 26	3syl 20	. . 3 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (rec(F, A) ‘suc (◡G ‘B)) = (F ‘(rec(F, A) ‘(◡G ‘B))))
28	6, 24, 27	3eqtrd 1514	. 2 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘(B + 1)) = (F ‘(rec(F, A) ‘(◡G ‘B))))
29	1, 4	uzrdgval 6303	. . 3 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘B) = (rec(F, A) ‘(◡G ‘B)))
30	29	fveq2d 3734	. 2 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → (F ‘((rec(F, A) ∘ ◡G) ‘B)) = (F ‘(rec(F, A) ‘(◡G ‘B))))
31	28, 30	eqtr4d 1513	1 ⊢ (B ∈ {z ∈ ℤ∣C ≤ z} → ((rec(F, A) ∘ ◡G) ‘(B + 1)) = (F ‘((rec(F, A) ∘ ◡G) ‘B)))

Syntax hints:   → wi 3   = wceq 958   ∈ wcel 960  {crab 1651   class class class wbr 2624  {copab 2671  Oncon0 2954  suc csuc 2956  ωcom 3137  ^◡ccnv 3175   ↾ cres 3178   ∘ ccom 3180  –→wf 3184  –1-1-onto→wf1o 3187   ‘cfv 3188  reccrdg 3937  (class class class)co 3969  1c1 5247   + caddc 5249   ≤ cle 5307  ℤcz 5310

This theorem is referenced by:  uzrdginip1 6306  seq1lem1 6310  seq1suclem 6317

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-n 5927  df-n0 6102  df-z 6138

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