Theorem repiecef 17295

Description: Piecewise definition on the reals yields a function. The function agrees with F and G on their respective parts of the real line; see repiecele0 17293 and repiecege0 17294. From an online post by James E Hanson. The construction was published in Martín Hötzel Escardó, "Effective and sequential definition by cases on the reals via infinite signed-digit numerals", Electronic Notes in Theoretical Computer Science 10 (1998), page 2, https://martinescardo.github.io/papers/lexnew.pdf. 17294 (Contributed by Jim Kingdon, 27-Apr-2026.)

Hypotheses

Ref	Expression
repiece.f	|- ( ph -> F : ( -oo (,] 0 ) --> RR )
repiece.g	|- ( ph -> G : ( 0 [,) +oo ) --> RR )
repiece.0	|- ( ph -> ( F ` 0 ) = ( G ` 0 ) )
repiece.h	|- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )

Assertion

Ref	Expression
repiecef	|- ( ph -> H : RR --> RR )

Distinct variable groups:   x, F   x, G

Allowed substitution hints:   ph( x)   H( x)

Proof of Theorem repiecef

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		repiece.f	. . . . 5 |- ( ph -> F : ( -oo (,] 0 ) --> RR )
2		repiece.g	. . . . 5 |- ( ph -> G : ( 0 [,) +oo ) --> RR )
3		repiece.0	. . . . 5 |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )
4		repiece.h	. . . . 5 |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )
5	1, 2, 3, 4	repiecelem 17292	. . . 4 |- ( ( ph /\ y e. RR ) -> ( ( ( F ` inf ( { y , 0 } , RR , < ) ) + ( G ` sup ( { y , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
6	5	ralrimiva 2623	. . 3 |- ( ph -> A. y e. RR ( ( ( F ` inf ( { y , 0 } , RR , < ) ) + ( G ` sup ( { y , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
7		preq1 3788	. . . . . . . . 9 |- ( y = x -> { y , 0 } = { x , 0 } )
8	7	infeq1d 7359	. . . . . . . 8 |- ( y = x -> inf ( { y , 0 } , RR , < ) = inf ( { x , 0 } , RR , < ) )
9	8	fveq2d 5702	. . . . . . 7 |- ( y = x -> ( F ` inf ( { y , 0 } , RR , < ) ) = ( F ` inf ( { x , 0 } , RR , < ) ) )
10	7	supeq1d 7334	. . . . . . . 8 |- ( y = x -> sup ( { y , 0 } , RR , < ) = sup ( { x , 0 } , RR , < ) )
11	10	fveq2d 5702	. . . . . . 7 |- ( y = x -> ( G ` sup ( { y , 0 } , RR , < ) ) = ( G ` sup ( { x , 0 } , RR , < ) ) )
12	9, 11	oveq12d 6106	. . . . . 6 |- ( y = x -> ( ( F ` inf ( { y , 0 } , RR , < ) ) + ( G ` sup ( { y , 0 } , RR , < ) ) ) = ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) )
13	12	oveq1d 6103	. . . . 5 |- ( y = x -> ( ( ( F ` inf ( { y , 0 } , RR , < ) ) + ( G ` sup ( { y , 0 } , RR , < ) ) ) - ( F ` 0 ) ) = ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )
14	13	eleq1d 2307	. . . 4 |- ( y = x -> ( ( ( ( F ` inf ( { y , 0 } , RR , < ) ) + ( G ` sup ( { y , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR <-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR ) )
15	14	cbvralv 2786	. . 3 |- ( A. y e. RR ( ( ( F ` inf ( { y , 0 } , RR , < ) ) + ( G ` sup ( { y , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR <-> A. x e. RR ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
16	6, 15	sylib 122	. 2 |- ( ph -> A. x e. RR ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
17	4	fmpt 5861	. 2 |- ( A. x e. RR ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR <-> H : RR --> RR )
18	16, 17	sylib 122	1 |- ( ph -> H : RR --> RR )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   {cpr 3710   |-> cmpt 4193   -->wf 5376   ` cfv 5380  (class class class)co 6088   supcsup 7329  infcinf 7330   RRcr 8185   0cc0 8186   + caddc 8189   +oocpnf 8364   -oocmnf 8365   < clt 8367   - cmin 8505   (,]cioc 10309   [,)cico 10310

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-rp 10073  df-ioc 10313  df-ico 10314  df-seqfrec 10907  df-exp 10998  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788

This theorem is used by: (None)