Theorem maxcncf 15131

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 15093	. . . . . 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 5740	. . . 4 |- ( ( ph /\ x e. X ) -> A e. RR )
5		maxcncf.b	. . . . . 6 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )
6		cncff 15093	. . . . . 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 5740	. . . 4 |- ( ( ph /\ x e. X ) -> B e. RR )
9		maxabs 11564	. . . 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 4138	. 2 |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) = ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) )
12	4, 8	readdcld 8109	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( A + B ) e. RR )
13	4, 8	resubcld 8460	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( A - B ) e. RR )
14	13	recnd 8108	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A - B ) e. CC )
15	14	abscld 11536	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( abs ` ( A - B ) ) e. RR )
16	12, 15	readdcld 8109	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
17	16	rehalfcld 9291	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
18	17	fmpttd 5742	. . 3 |- ( ph -> ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) : X --> RR )
19		ax-resscn 8024	. . . 4 |- RR C_ CC
20		ssid 3214	. . . . . . . . 9 |- CC C_ CC
21		cncfss 15099	. . . . . . . . 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 3192	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )
24	22, 5	sselid 3192	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )
25	23, 24	addcncf 15128	. . . . . 6 |- ( ph -> ( x e. X |-> ( A + B ) ) e. ( X -cn-> CC ) )
26		cncfss 15099	. . . . . . . . 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 15101	. . . . . . . . 9 |- abs e. ( CC -cn-> RR )
29	28	a1i 9	. . . . . . . 8 |- ( ph -> abs e. ( CC -cn-> RR ) )
30	27, 29	sselid 3192	. . . . . . 7 |- ( ph -> abs e. ( CC -cn-> CC ) )
31	23, 24	subcncf 15129	. . . . . . 7 |- ( ph -> ( x e. X |-> ( A - B ) ) e. ( X -cn-> CC ) )
32	30, 31	cncfmpt1f 15114	. . . . . 6 |- ( ph -> ( x e. X |-> ( abs ` ( A - B ) ) ) e. ( X -cn-> CC ) )
33	25, 32	addcncf 15128	. . . . 5 |- ( ph -> ( x e. X |-> ( ( A + B ) + ( abs ` ( A - B ) ) ) ) e. ( X -cn-> CC ) )
34		2cn 9114	. . . . . . 7 |- 2 e. CC
35		2ap0 9136	. . . . . . 7 |- 2 # 0
36		breq1 4050	. . . . . . . 8 |- ( y = 2 -> ( y # 0 <-> 2 # 0 ) )
37	36	elrab 2930	. . . . . . 7 |- ( 2 e. { y e. CC | y # 0 } <-> ( 2 e. CC /\ 2 # 0 ) )
38	34, 35, 37	mpbir2an 945	. . . . . 6 |- 2 e. { y e. CC | y # 0 }
39		cncfrss 15091	. . . . . . 7 |- ( ( x e. X |-> A ) e. ( X -cn-> RR ) -> X C_ CC )
40	1, 39	syl 14	. . . . . 6 |- ( ph -> X C_ CC )
41		apsscn 8727	. . . . . . 7 |- { y e. CC | y # 0 } C_ CC
42	41	a1i 9	. . . . . 6 |- ( ph -> { y e. CC | y # 0 } C_ CC )
43		cncfmptc 15112	. . . . . 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 1355	. . . . 5 |- ( ph -> ( x e. X |-> 2 ) e. ( X -cn-> { y e. CC | y # 0 } ) )
45	33, 44	divcncfap 15130	. . . 4 |- ( ph -> ( x e. X |-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) e. ( X -cn-> CC ) )
46		cncfcdm 15098	. . . 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 2283	1 |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   {crab 2489   C_ wss 3167   {cpr 3635   class class class wbr 4047   |-> cmpt 4109   -->wf 5272   ` cfv 5276  (class class class)co 5951   supcsup 7091   CCcc 7930   RRcr 7931   0cc0 7932   + caddc 7935   < clt 8114   - cmin 8250   # cap 8661   / cdiv 8752   2c2 9094   abscabs 11352   -cn->ccncf 15086

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052  ax-addf 8054

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-map 6744  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-xneg 9901  df-xadd 9902  df-seqfrec 10600  df-exp 10691  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-rest 13117  df-topgen 13136  df-psmet 14349  df-xmet 14350  df-met 14351  df-bl 14352  df-mopn 14353  df-top 14514  df-topon 14527  df-bases 14559  df-cn 14704  df-cnp 14705  df-tx 14769  df-cncf 15087

This theorem is referenced by:  hovercncf  15162