Theorem maxcncf 15342

Description: The maximum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 18-Jul-2025.)

Hypotheses

Ref	Expression
maxcncf.a	|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )
maxcncf.b	|- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )

Assertion

Ref	Expression
maxcncf	|- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )

Distinct variable groups:   x, X   ph, x

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem maxcncf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		maxcncf.a	. . . . . 6 |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )
2		cncff 15304	. . . . . 6 |- ( ( x e. X |-> A ) e. ( X -cn-> RR ) -> ( x e. X |-> A ) : X --> RR )
3	1, 2	syl 14	. . . . 5 |- ( ph -> ( x e. X |-> A ) : X --> RR )
4	3	fvmptelcdm 5800	. . . 4 |- ( ( ph /\ x e. X ) -> A e. RR )
5		maxcncf.b	. . . . . 6 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )
6		cncff 15304	. . . . . 6 |- ( ( x e. X |-> B ) e. ( X -cn-> RR ) -> ( x e. X |-> B ) : X --> RR )
7	5, 6	syl 14	. . . . 5 |- ( ph -> ( x e. X |-> B ) : X --> RR )
8	7	fvmptelcdm 5800	. . . 4 |- ( ( ph /\ x e. X ) -> B e. RR )
9		maxabs 11771	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
10	4, 8, 9	syl2anc 411	. . 3 |- ( ( ph /\ x e. X ) -> sup ( { A , B } , RR , < ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
11	10	mpteq2dva 4179	. 2 |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) = ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) )
12	4, 8	readdcld 8209	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( A + B ) e. RR )
13	4, 8	resubcld 8560	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( A - B ) e. RR )
14	13	recnd 8208	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A - B ) e. CC )
15	14	abscld 11743	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( abs ` ( A - B ) ) e. RR )
16	12, 15	readdcld 8209	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
17	16	rehalfcld 9391	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
18	17	fmpttd 5802	. . 3 |- ( ph -> ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) : X --> RR )
19		ax-resscn 8124	. . . 4 |- RR C_ CC
20		ssid 3247	. . . . . . . . 9 |- CC C_ CC
21		cncfss 15310	. . . . . . . . 9 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( X -cn-> RR ) C_ ( X -cn-> CC ) )
22	19, 20, 21	mp2an 426	. . . . . . . 8 |- ( X -cn-> RR ) C_ ( X -cn-> CC )
23	22, 1	sselid 3225	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )
24	22, 5	sselid 3225	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )
25	23, 24	addcncf 15339	. . . . . 6 |- ( ph -> ( x e. X |-> ( A + B ) ) e. ( X -cn-> CC ) )
26		cncfss 15310	. . . . . . . . 9 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( CC -cn-> RR ) C_ ( CC -cn-> CC ) )
27	19, 20, 26	mp2an 426	. . . . . . . 8 |- ( CC -cn-> RR ) C_ ( CC -cn-> CC )
28		abscncf 15312	. . . . . . . . 9 |- abs e. ( CC -cn-> RR )
29	28	a1i 9	. . . . . . . 8 |- ( ph -> abs e. ( CC -cn-> RR ) )
30	27, 29	sselid 3225	. . . . . . 7 |- ( ph -> abs e. ( CC -cn-> CC ) )
31	23, 24	subcncf 15340	. . . . . . 7 |- ( ph -> ( x e. X |-> ( A - B ) ) e. ( X -cn-> CC ) )
32	30, 31	cncfmpt1f 15325	. . . . . 6 |- ( ph -> ( x e. X |-> ( abs ` ( A - B ) ) ) e. ( X -cn-> CC ) )
33	25, 32	addcncf 15339	. . . . 5 |- ( ph -> ( x e. X |-> ( ( A + B ) + ( abs ` ( A - B ) ) ) ) e. ( X -cn-> CC ) )
34		2cn 9214	. . . . . . 7 |- 2 e. CC
35		2ap0 9236	. . . . . . 7 |- 2 # 0
36		breq1 4091	. . . . . . . 8 |- ( y = 2 -> ( y # 0 <-> 2 # 0 ) )
37	36	elrab 2962	. . . . . . 7 |- ( 2 e. { y e. CC | y # 0 } <-> ( 2 e. CC /\ 2 # 0 ) )
38	34, 35, 37	mpbir2an 950	. . . . . 6 |- 2 e. { y e. CC | y # 0 }
39		cncfrss 15302	. . . . . . 7 |- ( ( x e. X |-> A ) e. ( X -cn-> RR ) -> X C_ CC )
40	1, 39	syl 14	. . . . . 6 |- ( ph -> X C_ CC )
41		apsscn 8827	. . . . . . 7 |- { y e. CC | y # 0 } C_ CC
42	41	a1i 9	. . . . . 6 |- ( ph -> { y e. CC | y # 0 } C_ CC )
43		cncfmptc 15323	. . . . . 6 |- ( ( 2 e. { y e. CC | y # 0 } /\ X C_ CC /\ { y e. CC | y # 0 } C_ CC ) -> ( x e. X |-> 2 ) e. ( X -cn-> { y e. CC | y # 0 } ) )
44	38, 40, 42, 43	mp3an2i 1378	. . . . 5 |- ( ph -> ( x e. X |-> 2 ) e. ( X -cn-> { y e. CC | y # 0 } ) )
45	33, 44	divcncfap 15341	. . . 4 |- ( ph -> ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) e. ( X -cn-> CC ) )
46		cncfcdm 15309	. . . 4 |- ( ( RR C_ CC /\ ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) e. ( X -cn-> CC ) ) -> ( ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) e. ( X -cn-> RR ) <-> ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) : X --> RR ) )
47	19, 45, 46	sylancr 414	. . 3 |- ( ph -> ( ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) e. ( X -cn-> RR ) <-> ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) : X --> RR ) )
48	18, 47	mpbird 167	. 2 |- ( ph -> ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) e. ( X -cn-> RR ) )
49	11, 48	eqeltrd 2308	1 |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {crab 2514   C_ wss 3200   {cpr 3670   class class class wbr 4088   |-> cmpt 4150   -->wf 5322   ` cfv 5326  (class class class)co 6018   supcsup 7181   CCcc 8030   RRcr 8031   0cc0 8032   + caddc 8035   < clt 8214   - cmin 8350   # cap 8761   / cdiv 8852   2c2 9194   abscabs 11559   -cn->ccncf 15297

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  ax-addf 8154

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-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 11404  df-re 11405  df-im 11406  df-rsqrt 11560  df-abs 11561  df-rest 13326  df-topgen 13345  df-psmet 14560  df-xmet 14561  df-met 14562  df-bl 14563  df-mopn 14564  df-top 14725  df-topon 14738  df-bases 14770  df-cn 14915  df-cnp 14916  df-tx 14980  df-cncf 15298

This theorem is referenced by:  hovercncf  15373