Theorem maxcncf 15254

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 15216	. . . . . 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 5761	. . . 4 |- ( ( ph /\ x e. X ) -> A e. RR )
5		maxcncf.b	. . . . . 6 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )
6		cncff 15216	. . . . . 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 5761	. . . 4 |- ( ( ph /\ x e. X ) -> B e. RR )
9		maxabs 11686	. . . 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 4153	. 2 |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) = ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) )
12	4, 8	readdcld 8144	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( A + B ) e. RR )
13	4, 8	resubcld 8495	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( A - B ) e. RR )
14	13	recnd 8143	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A - B ) e. CC )
15	14	abscld 11658	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( abs ` ( A - B ) ) e. RR )
16	12, 15	readdcld 8144	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
17	16	rehalfcld 9326	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
18	17	fmpttd 5763	. . 3 |- ( ph -> ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) : X --> RR )
19		ax-resscn 8059	. . . 4 |- RR C_ CC
20		ssid 3224	. . . . . . . . 9 |- CC C_ CC
21		cncfss 15222	. . . . . . . . 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 3202	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )
24	22, 5	sselid 3202	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )
25	23, 24	addcncf 15251	. . . . . 6 |- ( ph -> ( x e. X |-> ( A + B ) ) e. ( X -cn-> CC ) )
26		cncfss 15222	. . . . . . . . 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 15224	. . . . . . . . 9 |- abs e. ( CC -cn-> RR )
29	28	a1i 9	. . . . . . . 8 |- ( ph -> abs e. ( CC -cn-> RR ) )
30	27, 29	sselid 3202	. . . . . . 7 |- ( ph -> abs e. ( CC -cn-> CC ) )
31	23, 24	subcncf 15252	. . . . . . 7 |- ( ph -> ( x e. X |-> ( A - B ) ) e. ( X -cn-> CC ) )
32	30, 31	cncfmpt1f 15237	. . . . . 6 |- ( ph -> ( x e. X |-> ( abs ` ( A - B ) ) ) e. ( X -cn-> CC ) )
33	25, 32	addcncf 15251	. . . . 5 |- ( ph -> ( x e. X |-> ( ( A + B ) + ( abs ` ( A - B ) ) ) ) e. ( X -cn-> CC ) )
34		2cn 9149	. . . . . . 7 |- 2 e. CC
35		2ap0 9171	. . . . . . 7 |- 2 # 0
36		breq1 4065	. . . . . . . 8 |- ( y = 2 -> ( y # 0 <-> 2 # 0 ) )
37	36	elrab 2939	. . . . . . 7 |- ( 2 e. { y e. CC | y # 0 } <-> ( 2 e. CC /\ 2 # 0 ) )
38	34, 35, 37	mpbir2an 947	. . . . . 6 |- 2 e. { y e. CC | y # 0 }
39		cncfrss 15214	. . . . . . 7 |- ( ( x e. X |-> A ) e. ( X -cn-> RR ) -> X C_ CC )
40	1, 39	syl 14	. . . . . 6 |- ( ph -> X C_ CC )
41		apsscn 8762	. . . . . . 7 |- { y e. CC | y # 0 } C_ CC
42	41	a1i 9	. . . . . 6 |- ( ph -> { y e. CC | y # 0 } C_ CC )
43		cncfmptc 15235	. . . . . 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 1357	. . . . 5 |- ( ph -> ( x e. X |-> 2 ) e. ( X -cn-> { y e. CC | y # 0 } ) )
45	33, 44	divcncfap 15253	. . . 4 |- ( ph -> ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) e. ( X -cn-> CC ) )
46		cncfcdm 15221	. . . 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 2286	1 |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   e. wcel 2180   {crab 2492   C_ wss 3177   {cpr 3647   class class class wbr 4062   |-> cmpt 4124   -->wf 5290   ` cfv 5294  (class class class)co 5974   supcsup 7117   CCcc 7965   RRcr 7966   0cc0 7967   + caddc 7970   < clt 8149   - cmin 8285   # cap 8696   / cdiv 8787   2c2 9129   abscabs 11474   -cn->ccncf 15209

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087  ax-addf 8089

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-map 6767  df-sup 7119  df-inf 7120  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-n0 9338  df-z 9415  df-uz 9691  df-q 9783  df-rp 9818  df-xneg 9936  df-xadd 9937  df-seqfrec 10637  df-exp 10728  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476  df-rest 13240  df-topgen 13259  df-psmet 14472  df-xmet 14473  df-met 14474  df-bl 14475  df-mopn 14476  df-top 14637  df-topon 14650  df-bases 14682  df-cn 14827  df-cnp 14828  df-tx 14892  df-cncf 15210

This theorem is referenced by:  hovercncf  15285