Theorem dvidsslem 15484

Description: Lemma for dvconstss 15489. Analogue of dvidlemap 15482 where F is defined on an open subset of the real or complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)

Hypotheses

Ref	Expression
dvidsslem.s	|- ( ph -> S e. { RR , CC } )
dvidsslem.j	|- J = ( K ↾t S )
dvidsslem.k	|- K = ( MetOpen ` ( abs o. - ) )
dvidsslem.1	|- ( ph -> F : X --> CC )
dvidsslem.x	|- ( ph -> X e. J )
dvidsslem.2	|- ( ( ph /\ ( x e. X /\ z e. X /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
dvidsslem.3	|- B e. CC

Assertion

Ref	Expression
dvidsslem	|- ( ph -> ( S _D F ) = ( X X. { B } ) )

Distinct variable groups:   x, z, B   x, F, z   ph, x, z   x, S, z   x, X, z

Allowed substitution hints:   J( x, z)   K( x, z)

Proof of Theorem dvidsslem

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvidsslem.s	. . . . 5 |- ( ph -> S e. { RR , CC } )
2		ssidd 3249	. . . . . . 7 |- ( ph -> CC C_ CC )
3		dvidsslem.j	. . . . . . . . . 10 |- J = ( K ↾t S )
4		restsspw 13393	. . . . . . . . . 10 |- ( K ↾t S ) C_ ~P S
5	3, 4	eqsstri 3260	. . . . . . . . 9 |- J C_ ~P S
6		dvidsslem.x	. . . . . . . . 9 |- ( ph -> X e. J )
7	5, 6	sselid 3226	. . . . . . . 8 |- ( ph -> X e. ~P S )
8	7	elpwid 3667	. . . . . . 7 |- ( ph -> X C_ S )
9		cnex 8199	. . . . . . . 8 |- CC e. _V
10	9	a1i 9	. . . . . . 7 |- ( ph -> CC e. _V )
11		pmss12g 6887	. . . . . . 7 |- ( ( ( CC C_ CC /\ X C_ S ) /\ ( CC e. _V /\ S e. { RR , CC } ) ) -> ( CC ^pm X ) C_ ( CC ^pm S ) )
12	2, 8, 10, 1, 11	syl22anc 1275	. . . . . 6 |- ( ph -> ( CC ^pm X ) C_ ( CC ^pm S ) )
13		dvidsslem.1	. . . . . . 7 |- ( ph -> F : X --> CC )
14		fpmg 6886	. . . . . . 7 |- ( ( X e. J /\ CC e. _V /\ F : X --> CC ) -> F e. ( CC ^pm X ) )
15	6, 10, 13, 14	syl3anc 1274	. . . . . 6 |- ( ph -> F e. ( CC ^pm X ) )
16	12, 15	sseldd 3229	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
17		dvfgg 15479	. . . . 5 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
18	1, 16, 17	syl2anc 411	. . . 4 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
19		recnprss 15478	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
20	1, 19	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
21	20, 13, 8	dvbss 15476	. . . . . 6 |- ( ph -> dom ( S _D F ) C_ X )
22		reldvg 15470	. . . . . . . . 9 |- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> Rel ( S _D F ) )
23	20, 16, 22	syl2anc 411	. . . . . . . 8 |- ( ph -> Rel ( S _D F ) )
24	23	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. X ) -> Rel ( S _D F ) )
25		dvidsslem.k	. . . . . . . . . . . . . . . 16 |- K = ( MetOpen ` ( abs o. - ) )
26	25	cntoptop 15324	. . . . . . . . . . . . . . 15 |- K e. Top
27	26	a1i 9	. . . . . . . . . . . . . 14 |- ( ph -> K e. Top )
28		resttop 14961	. . . . . . . . . . . . . 14 |- ( ( K e. Top /\ S e. { RR , CC } ) -> ( K ↾t S ) e. Top )
29	27, 1, 28	syl2anc 411	. . . . . . . . . . . . 13 |- ( ph -> ( K ↾t S ) e. Top )
30	3, 29	eqeltrid 2318	. . . . . . . . . . . 12 |- ( ph -> J e. Top )
31		isopn3i 14926	. . . . . . . . . . . 12 |- ( ( J e. Top /\ X e. J ) -> ( ( int ` J ) ` X ) = X )
32	30, 6, 31	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( ( int ` J ) ` X ) = X )
33	32	eqcomd 2237	. . . . . . . . . 10 |- ( ph -> X = ( ( int ` J ) ` X ) )
34	33	eleq2d 2301	. . . . . . . . 9 |- ( ph -> ( x e. X <-> x e. ( ( int ` J ) ` X ) ) )
35	34	biimpa 296	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x e. ( ( int ` J ) ` X ) )
36		limcresi 15457	. . . . . . . . . 10 |- ( ( z e. X |-> B ) lim CC x ) C_ ( ( ( z e. X |-> B ) |` { w e. X | w # x } ) lim CC x )
37		dvidsslem.3	. . . . . . . . . . . . . 14 |- B e. CC
38	37	a1i 9	. . . . . . . . . . . . 13 |- ( ph -> B e. CC )
39	8, 20	sstrd 3238	. . . . . . . . . . . . 13 |- ( ph -> X C_ CC )
40		cncfmptc 15387	. . . . . . . . . . . . 13 |- ( ( B e. CC /\ X C_ CC /\ CC C_ CC ) -> ( z e. X |-> B ) e. ( X -cn-> CC ) )
41	38, 39, 2, 40	syl3anc 1274	. . . . . . . . . . . 12 |- ( ph -> ( z e. X |-> B ) e. ( X -cn-> CC ) )
42	41	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( z e. X |-> B ) e. ( X -cn-> CC ) )
43		simpr 110	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> x e. X )
44		eqidd 2232	. . . . . . . . . . 11 |- ( z = x -> B = B )
45	42, 43, 44	cnmptlimc 15465	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> B e. ( ( z e. X |-> B ) lim CC x ) )
46	36, 45	sselid 3226	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> B e. ( ( ( z e. X |-> B ) |` { w e. X | w # x } ) lim CC x ) )
47		breq1 4096	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
48	47	elrab 2963	. . . . . . . . . . . . 13 |- ( z e. { w e. X | w # x } <-> ( z e. X /\ z # x ) )
49		dvidsslem.2	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x e. X /\ z e. X /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
50	49	3exp2 1252	. . . . . . . . . . . . . 14 |- ( ph -> ( x e. X -> ( z e. X -> ( z # x -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B ) ) ) )
51	50	imp43 355	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. X ) /\ ( z e. X /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
52	48, 51	sylan2b 287	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. X ) /\ z e. { w e. X | w # x } ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )
53	52	mpteq2dva 4184	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. X | w # x } |-> B ) )
54		ssrab2 3313	. . . . . . . . . . . 12 |- { w e. X | w # x } C_ X
55		resmpt 5067	. . . . . . . . . . . 12 |- ( { w e. X | w # x } C_ X -> ( ( z e. X |-> B ) |` { w e. X | w # x } ) = ( z e. { w e. X | w # x } |-> B ) )
56	54, 55	ax-mp 5	. . . . . . . . . . 11 |- ( ( z e. X |-> B ) |` { w e. X | w # x } ) = ( z e. { w e. X | w # x } |-> B )
57	53, 56	eqtr4di 2282	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( ( z e. X |-> B ) |` { w e. X | w # x } ) )
58	57	oveq1d 6043	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) = ( ( ( z e. X |-> B ) |` { w e. X | w # x } ) lim CC x ) )
59	46, 58	eleqtrrd 2311	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> B e. ( ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) )
60		eqid 2231	. . . . . . . . . 10 |- ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) = ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) )
61	3, 25, 60, 20, 13, 8	eldvap 15473	. . . . . . . . 9 |- ( ph -> ( x ( S _D F ) B <-> ( x e. ( ( int ` J ) ` X ) /\ B e. ( ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
62	61	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( x ( S _D F ) B <-> ( x e. ( ( int ` J ) ` X ) /\ B e. ( ( z e. { w e. X | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) ) )
63	35, 59, 62	mpbir2and 953	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D F ) B )
64		releldm 4973	. . . . . . 7 |- ( ( Rel ( S _D F ) /\ x ( S _D F ) B ) -> x e. dom ( S _D F ) )
65	24, 63, 64	syl2anc 411	. . . . . 6 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D F ) )
66	21, 65	eqelssd 3247	. . . . 5 |- ( ph -> dom ( S _D F ) = X )
67	66	feq2d 5477	. . . 4 |- ( ph -> ( ( S _D F ) : dom ( S _D F ) --> CC <-> ( S _D F ) : X --> CC ) )
68	18, 67	mpbid 147	. . 3 |- ( ph -> ( S _D F ) : X --> CC )
69	68	ffnd 5490	. 2 |- ( ph -> ( S _D F ) Fn X )
70		fnconstg 5543	. . 3 |- ( B e. CC -> ( X X. { B } ) Fn X )
71	37, 70	mp1i 10	. 2 |- ( ph -> ( X X. { B } ) Fn X )
72	18	adantr 276	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( S _D F ) : dom ( S _D F ) --> CC )
73	72	ffund 5493	. . . . 5 |- ( ( ph /\ x e. X ) -> Fun ( S _D F ) )
74		funbrfvb 5695	. . . . 5 |- ( ( Fun ( S _D F ) /\ x e. dom ( S _D F ) ) -> ( ( ( S _D F ) ` x ) = B <-> x ( S _D F ) B ) )
75	73, 65, 74	syl2anc 411	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) = B <-> x ( S _D F ) B ) )
76	63, 75	mpbird 167	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = B )
77		fvconst2g 5876	. . . 4 |- ( ( B e. CC /\ x e. X ) -> ( ( X X. { B } ) ` x ) = B )
78	38, 77	sylan 283	. . 3 |- ( ( ph /\ x e. X ) -> ( ( X X. { B } ) ` x ) = B )
79	76, 78	eqtr4d 2267	. 2 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) = ( ( X X. { B } ) ` x ) )
80	69, 71, 79	eqfnfvd 5756	1 |- ( ph -> ( S _D F ) = ( X X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   {crab 2515   _Vcvv 2803   C_ wss 3201   ~Pcpw 3656   {csn 3673   {cpr 3674   class class class wbr 4093   |-> cmpt 4155   X. cxp 4729   dom cdm 4731   |` cres 4733   o. ccom 4735   Rel wrel 4736   Fun wfun 5327   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   ^pm cpm 6861   CCcc 8073   RRcr 8074   - cmin 8393   # cap 8804   / cdiv 8895   abscabs 11618   ↾_t crest 13383   MetOpencmopn 14617   Topctop 14788   intcnt 14884   -cn->ccncf 15361   lim CC climc 15445   _D cdv 15446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-map 6862  df-pm 6863  df-sup 7226  df-inf 7227  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-n0 9446  df-z 9523  df-uz 9799  df-q 9897  df-rp 9932  df-xneg 10050  df-xadd 10051  df-seqfrec 10754  df-exp 10845  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-rest 13385  df-topgen 13404  df-psmet 14619  df-xmet 14620  df-met 14621  df-bl 14622  df-mopn 14623  df-top 14789  df-topon 14802  df-bases 14834  df-ntr 14887  df-cn 14979  df-cnp 14980  df-cncf 15362  df-limced 15447  df-dvap 15448

This theorem is referenced by:  dvconstss  15489