Theorem fvmptnn04if 19872

Description: The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)

Hypotheses

Ref	Expression
fvmptnn04if.g	|- G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
fvmptnn04if.s	|- ( ph -> S e. NN )
fvmptnn04if.n	|- ( ph -> N e. NN0 )
fvmptnn04if.y	|- ( ph -> Y e. V )
fvmptnn04if.a	|- ( ( ph /\ N = 0 ) -> Y = [_ N / n ]_ A )
fvmptnn04if.b	|- ( ( ph /\ 0 < N /\ N < S ) -> Y = [_ N / n ]_ B )
fvmptnn04if.c	|- ( ( ph /\ N = S ) -> Y = [_ N / n ]_ C )
fvmptnn04if.d	|- ( ( ph /\ S < N ) -> Y = [_ N / n ]_ D )

Assertion

Ref	Expression
fvmptnn04if	|- ( ph -> ( G ` N ) = Y )

Distinct variable groups:   n, N   S, n

Allowed substitution hints:   ph( n)   A( n)   B( n)   C( n)   D( n)   G( n)   V( n)   Y( n)

Proof of Theorem fvmptnn04if

Step	Hyp	Ref	Expression
1		fvmptnn04if.n	. . 3 |- ( ph -> N e. NN0 )
2		csbif 3961	. . . . 5 |- [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) = if ( [. N / n ]. n = 0 , [_ N / n ]_ A , [_ N / n ]_ if ( n = S , C , if ( S < n , D , B ) ) )
3		eqsbc3 3339	. . . . . . 7 |- ( N e. NN0 -> ( [. N / n ]. n = 0 <-> N = 0 ) )
4	1, 3	syl 17	. . . . . 6 |- ( ph -> ( [. N / n ]. n = 0 <-> N = 0 ) )
5		csbif 3961	. . . . . . 7 |- [_ N / n ]_ if ( n = S , C , if ( S < n , D , B ) ) = if ( [. N / n ]. n = S , [_ N / n ]_ C , [_ N / n ]_ if ( S < n , D , B ) )
6		eqsbc3 3339	. . . . . . . . 9 |- ( N e. NN0 -> ( [. N / n ]. n = S <-> N = S ) )
7	1, 6	syl 17	. . . . . . . 8 |- ( ph -> ( [. N / n ]. n = S <-> N = S ) )
8		csbif 3961	. . . . . . . . 9 |- [_ N / n ]_ if ( S < n , D , B ) = if ( [. N / n ]. S < n , [_ N / n ]_ D , [_ N / n ]_ B )
9		sbcbr2g 4479	. . . . . . . . . . . 12 |- ( N e. NN0 -> ( [. N / n ]. S < n <-> S < [_ N / n ]_ n ) )
10	1, 9	syl 17	. . . . . . . . . . 11 |- ( ph -> ( [. N / n ]. S < n <-> S < [_ N / n ]_ n ) )
11		csbvarg 3822	. . . . . . . . . . . . 13 |- ( N e. NN0 -> [_ N / n ]_ n = N )
12	1, 11	syl 17	. . . . . . . . . . . 12 |- ( ph -> [_ N / n ]_ n = N )
13	12	breq2d 4435	. . . . . . . . . . 11 |- ( ph -> ( S < [_ N / n ]_ n <-> S < N ) )
14	10, 13	bitrd 256	. . . . . . . . . 10 |- ( ph -> ( [. N / n ]. S < n <-> S < N ) )
15	14	ifbid 3933	. . . . . . . . 9 |- ( ph -> if ( [. N / n ]. S < n , [_ N / n ]_ D , [_ N / n ]_ B ) = if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) )
16	8, 15	syl5eq 2475	. . . . . . . 8 |- ( ph -> [_ N / n ]_ if ( S < n , D , B ) = if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) )
17	7, 16	ifbieq2d 3936	. . . . . . 7 |- ( ph -> if ( [. N / n ]. n = S , [_ N / n ]_ C , [_ N / n ]_ if ( S < n , D , B ) ) = if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) )
18	5, 17	syl5eq 2475	. . . . . 6 |- ( ph -> [_ N / n ]_ if ( n = S , C , if ( S < n , D , B ) ) = if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) )
19	4, 18	ifbieq2d 3936	. . . . 5 |- ( ph -> if ( [. N / n ]. n = 0 , [_ N / n ]_ A , [_ N / n ]_ if ( n = S , C , if ( S < n , D , B ) ) ) = if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) )
20	2, 19	syl5eq 2475	. . . 4 |- ( ph -> [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) = if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) )
21		fvmptnn04if.a	. . . . . 6 |- ( ( ph /\ N = 0 ) -> Y = [_ N / n ]_ A )
22		fvmptnn04if.y	. . . . . . 7 |- ( ph -> Y e. V )
23	22	adantr 466	. . . . . 6 |- ( ( ph /\ N = 0 ) -> Y e. V )
24	21, 23	eqeltrrd 2508	. . . . 5 |- ( ( ph /\ N = 0 ) -> [_ N / n ]_ A e. V )
25		fvmptnn04if.c	. . . . . . . . 9 |- ( ( ph /\ N = S ) -> Y = [_ N / n ]_ C )
26	25	eqcomd 2430	. . . . . . . 8 |- ( ( ph /\ N = S ) -> [_ N / n ]_ C = Y )
27	26	adantlr 719	. . . . . . 7 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> [_ N / n ]_ C = Y )
28	22	ad2antrr 730	. . . . . . 7 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> Y e. V )
29	27, 28	eqeltrd 2507	. . . . . 6 |- ( ( ( ph /\ -. N = 0 ) /\ N = S ) -> [_ N / n ]_ C e. V )
30		fvmptnn04if.d	. . . . . . . . . . . 12 |- ( ( ph /\ S < N ) -> Y = [_ N / n ]_ D )
31	30	eqcomd 2430	. . . . . . . . . . 11 |- ( ( ph /\ S < N ) -> [_ N / n ]_ D = Y )
32	31	ex 435	. . . . . . . . . 10 |- ( ph -> ( S < N -> [_ N / n ]_ D = Y ) )
33	32	ad2antrr 730	. . . . . . . . 9 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> ( S < N -> [_ N / n ]_ D = Y ) )
34	33	imp 430	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D = Y )
35	22	ad3antrrr 734	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> Y e. V )
36	34, 35	eqeltrd 2507	. . . . . . 7 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ S < N ) -> [_ N / n ]_ D e. V )
37		simplll 766	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> ph )
38		ancom 451	. . . . . . . . . . . . 13 |- ( ( -. S < N /\ ph ) <-> ( ph /\ -. S < N ) )
39	38	anbi2i 698	. . . . . . . . . . . 12 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( ph /\ -. S < N ) ) )
40		ancom 451	. . . . . . . . . . . . 13 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) /\ ph ) )
41		anass 653	. . . . . . . . . . . . . . 15 |- ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) <-> ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) )
42	41	bicomi 205	. . . . . . . . . . . . . 14 |- ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) )
43	42	anbi1i 699	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) /\ ph ) <-> ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) /\ ph ) )
44		anass 653	. . . . . . . . . . . . 13 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ -. S < N ) /\ ph ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) )
45	40, 43, 44	3bitri 274	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( -. S < N /\ ph ) ) )
46		anass 653	. . . . . . . . . . . 12 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) <-> ( ( -. N = 0 /\ -. N = S ) /\ ( ph /\ -. S < N ) ) )
47	39, 45, 46	3bitr4i 280	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) )
48		an32 805	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( -. N = 0 /\ ph ) /\ -. N = S ) )
49		ancom 451	. . . . . . . . . . . . . 14 |- ( ( -. N = 0 /\ ph ) <-> ( ph /\ -. N = 0 ) )
50	49	anbi1i 699	. . . . . . . . . . . . 13 |- ( ( ( -. N = 0 /\ ph ) /\ -. N = S ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
51	48, 50	bitri 252	. . . . . . . . . . . 12 |- ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) <-> ( ( ph /\ -. N = 0 ) /\ -. N = S ) )
52	51	anbi1i 699	. . . . . . . . . . 11 |- ( ( ( ( -. N = 0 /\ -. N = S ) /\ ph ) /\ -. S < N ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
53	47, 52	bitri 252	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) <-> ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) )
54		df-ne 2616	. . . . . . . . . . . . 13 |- ( N =/= 0 <-> -. N = 0 )
55		elnnne0 10891	. . . . . . . . . . . . . . 15 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
56		nngt0 10646	. . . . . . . . . . . . . . 15 |- ( N e. NN -> 0 < N )
57	55, 56	sylbir 216	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ N =/= 0 ) -> 0 < N )
58	57	expcom 436	. . . . . . . . . . . . 13 |- ( N =/= 0 -> ( N e. NN0 -> 0 < N ) )
59	54, 58	sylbir 216	. . . . . . . . . . . 12 |- ( -. N = 0 -> ( N e. NN0 -> 0 < N ) )
60	59	adantr 466	. . . . . . . . . . 11 |- ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> ( N e. NN0 -> 0 < N ) )
61	1, 60	mpan9 471	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> 0 < N )
62	53, 61	sylbir 216	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> 0 < N )
63	1	nn0red 10934	. . . . . . . . . . . . . . . 16 |- ( ph -> N e. RR )
64		fvmptnn04if.s	. . . . . . . . . . . . . . . . 17 |- ( ph -> S e. NN )
65	64	nnred 10632	. . . . . . . . . . . . . . . 16 |- ( ph -> S e. RR )
66	63, 65	lenltd 9789	. . . . . . . . . . . . . . 15 |- ( ph -> ( N <_ S <-> -. S < N ) )
67	66	biimprd 226	. . . . . . . . . . . . . 14 |- ( ph -> ( -. S < N -> N <_ S ) )
68	67	adantld 468	. . . . . . . . . . . . 13 |- ( ph -> ( ( -. N = S /\ -. S < N ) -> N <_ S ) )
69	68	adantld 468	. . . . . . . . . . . 12 |- ( ph -> ( ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) -> N <_ S ) )
70	69	imp 430	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N <_ S )
71		nesym 2692	. . . . . . . . . . . . . 14 |- ( S =/= N <-> -. N = S )
72	71	biimpri 209	. . . . . . . . . . . . 13 |- ( -. N = S -> S =/= N )
73	72	adantr 466	. . . . . . . . . . . 12 |- ( ( -. N = S /\ -. S < N ) -> S =/= N )
74	73	ad2antll 733	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S =/= N )
75	63	adantr 466	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N e. RR )
76	65	adantr 466	. . . . . . . . . . . 12 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> S e. RR )
77	75, 76	ltlend 9788	. . . . . . . . . . 11 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> ( N < S <-> ( N <_ S /\ S =/= N ) ) )
78	70, 74, 77	mpbir2and 930	. . . . . . . . . 10 |- ( ( ph /\ ( -. N = 0 /\ ( -. N = S /\ -. S < N ) ) ) -> N < S )
79	53, 78	sylbir 216	. . . . . . . . 9 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> N < S )
80		fvmptnn04if.b	. . . . . . . . . 10 |- ( ( ph /\ 0 < N /\ N < S ) -> Y = [_ N / n ]_ B )
81	80	eqcomd 2430	. . . . . . . . 9 |- ( ( ph /\ 0 < N /\ N < S ) -> [_ N / n ]_ B = Y )
82	37, 62, 79, 81	syl3anc 1264	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B = Y )
83	22	ad3antrrr 734	. . . . . . . 8 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> Y e. V )
84	82, 83	eqeltrd 2507	. . . . . . 7 |- ( ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) /\ -. S < N ) -> [_ N / n ]_ B e. V )
85	36, 84	ifclda 3943	. . . . . 6 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) e. V )
86	29, 85	ifclda 3943	. . . . 5 |- ( ( ph /\ -. N = 0 ) -> if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) e. V )
87	24, 86	ifclda 3943	. . . 4 |- ( ph -> if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) e. V )
88	20, 87	eqeltrd 2507	. . 3 |- ( ph -> [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) e. V )
89		fvmptnn04if.g	. . . 4 |- G = ( n e. NN0 |-> if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
90	89	fvmpts 5968	. . 3 |- ( ( N e. NN0 /\ [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) e. V ) -> ( G ` N ) = [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
91	1, 88, 90	syl2anc 665	. 2 |- ( ph -> ( G ` N ) = [_ N / n ]_ if ( n = 0 , A , if ( n = S , C , if ( S < n , D , B ) ) ) )
92	21	eqcomd 2430	. . 3 |- ( ( ph /\ N = 0 ) -> [_ N / n ]_ A = Y )
93	34, 82	ifeqda 3944	. . . 4 |- ( ( ( ph /\ -. N = 0 ) /\ -. N = S ) -> if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) = Y )
94	27, 93	ifeqda 3944	. . 3 |- ( ( ph /\ -. N = 0 ) -> if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) = Y )
95	92, 94	ifeqda 3944	. 2 |- ( ph -> if ( N = 0 , [_ N / n ]_ A , if ( N = S , [_ N / n ]_ C , if ( S < N , [_ N / n ]_ D , [_ N / n ]_ B ) ) ) = Y )
96	91, 20, 95	3eqtrd 2467	1 |- ( ph -> ( G ` N ) = Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   =/= wne 2614   [.wsbc 3299   [_csb 3395   ifcif 3911   class class class wbr 4423   |-> cmpt 4482   ` cfv 5601   RRcr 9546   0cc0 9547   < clt 9683   <_ cle 9684   NNcn 10617   NN0cn0 10877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-n0 10878

This theorem is referenced by:  fvmptnn04ifa  19873  fvmptnn04ifb  19874  fvmptnn04ifc  19875  fvmptnn04ifd  19876