Theorem dvidsslem 15423

Description: Lemma for dvconstss 15428. Analogue of dvidlemap 15421 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 3248	. . . . . . 7 |- ( ph -> CC C_ CC )
3		dvidsslem.j	. . . . . . . . . 10 |- J = ( K ↾t S )
4		restsspw 13337	. . . . . . . . . 10 |- ( K ↾t S ) C_ ~P S
5	3, 4	eqsstri 3259	. . . . . . . . 9 |- J C_ ~P S
6		dvidsslem.x	. . . . . . . . 9 |- ( ph -> X e. J )
7	5, 6	sselid 3225	. . . . . . . 8 |- ( ph -> X e. ~P S )
8	7	elpwid 3663	. . . . . . 7 |- ( ph -> X C_ S )
9		cnex 8156	. . . . . . . 8 |- CC e. _V
10	9	a1i 9	. . . . . . 7 |- ( ph -> CC e. _V )
11		pmss12g 6844	. . . . . . 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 1274	. . . . . 6 |- ( ph -> ( CC ^pm X ) C_ ( CC ^pm S ) )
13		dvidsslem.1	. . . . . . 7 |- ( ph -> F : X --> CC )
14		fpmg 6843	. . . . . . 7 |- ( ( X e. J /\ CC e. _V /\ F : X --> CC ) -> F e. ( CC ^pm X ) )
15	6, 10, 13, 14	syl3anc 1273	. . . . . 6 |- ( ph -> F e. ( CC ^pm X ) )
16	12, 15	sseldd 3228	. . . . 5 |- ( ph -> F e. ( CC ^pm S ) )
17		dvfgg 15418	. . . . 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 15417	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
20	1, 19	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
21	20, 13, 8	dvbss 15415	. . . . . 6 |- ( ph -> dom ( S _D F ) C_ X )
22		reldvg 15409	. . . . . . . . 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 15263	. . . . . . . . . . . . . . 15 |- K e. Top
27	26	a1i 9	. . . . . . . . . . . . . 14 |- ( ph -> K e. Top )
28		resttop 14900	. . . . . . . . . . . . . 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 14865	. . . . . . . . . . . 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 15396	. . . . . . . . . 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 3237	. . . . . . . . . . . . 13 |- ( ph -> X C_ CC )
40		cncfmptc 15326	. . . . . . . . . . . . 13 |- ( ( B e. CC /\ X C_ CC /\ CC C_ CC ) -> ( z e. X |-> B ) e. ( X -cn-> CC ) )
41	38, 39, 2, 40	syl3anc 1273	. . . . . . . . . . . 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 15404	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> B e. ( ( z e. X |-> B ) lim CC x ) )
46	36, 45	sselid 3225	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> B e. ( ( ( z e. X |-> B ) |` { w e. X | w # x } ) lim CC x ) )
47		breq1 4091	. . . . . . . . . . . . . 14 |- ( w = z -> ( w # x <-> z # x ) )
48	47	elrab 2962	. . . . . . . . . . . . 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 1251	. . . . . . . . . . . . . 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 4179	. . . . . . . . . . 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 3312	. . . . . . . . . . . 12 |- { w e. X | w # x } C_ X
55		resmpt 5061	. . . . . . . . . . . 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 6033	. . . . . . . . 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 15412	. . . . . . . . 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 952	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D F ) B )
64		releldm 4967	. . . . . . 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 3246	. . . . 5 |- ( ph -> dom ( S _D F ) = X )
67	66	feq2d 5470	. . . 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 5483	. 2 |- ( ph -> ( S _D F ) Fn X )
70		fnconstg 5534	. . 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 5486	. . . . 5 |- ( ( ph /\ x e. X ) -> Fun ( S _D F ) )
74		funbrfvb 5686	. . . . 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 5868	. . . 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 5747	1 |- ( ph -> ( S _D F ) = ( X X. { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   {csn 3669   {cpr 3670   class class class wbr 4088   |-> cmpt 4150   X. cxp 4723   dom cdm 4725   |` cres 4727   o. ccom 4729   Rel wrel 4730   Fun wfun 5320   Fn wfn 5321   -->wf 5322   ` cfv 5326  (class class class)co 6018   ^pm cpm 6818   CCcc 8030   RRcr 8031   - cmin 8350   # cap 8761   / cdiv 8852   abscabs 11562   ↾_t crest 13327   MetOpencmopn 14561   Topctop 14727   intcnt 14823   -cn->ccncf 15300   lim CC climc 15384   _D cdv 15385

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-pm 6820  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-rest 13329  df-topgen 13348  df-psmet 14563  df-xmet 14564  df-met 14565  df-bl 14566  df-mopn 14567  df-top 14728  df-topon 14741  df-bases 14773  df-ntr 14826  df-cn 14918  df-cnp 14919  df-cncf 15301  df-limced 15386  df-dvap 15387

This theorem is referenced by:  dvconstss  15428