Theorem mincncf 14936

Description: The minimum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 19-Jul-2025.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x, X   ph, x

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

Proof of Theorem mincncf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mincncf.a	. . . . . 6 |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )
2		cncff 14897	. . . . . 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 5718	. . . 4 |- ( ( ph /\ x e. X ) -> A e. RR )
5		mincncf.b	. . . . . 6 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )
6		cncff 14897	. . . . . 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 5718	. . . 4 |- ( ( ph /\ x e. X ) -> B e. RR )
9		minabs 11418	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) )
10	4, 8, 9	syl2anc 411	. . 3 |- ( ( ph /\ x e. X ) -> inf ( { A , B } , RR , < ) = ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) )
11	10	mpteq2dva 4124	. 2 |- ( ph -> ( x e. X |-> inf ( { A , B } , RR , < ) ) = ( x e. X |-> ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) ) )
12	4, 8	readdcld 8073	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( A + B ) e. RR )
13	4, 8	resubcld 8424	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( A - B ) e. RR )
14	13	recnd 8072	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A - B ) e. CC )
15	14	abscld 11363	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( abs ` ( A - B ) ) e. RR )
16	12, 15	resubcld 8424	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A + B ) - ( abs ` ( A - B ) ) ) e. RR )
17	16	rehalfcld 9255	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) e. RR )
18	17	fmpttd 5720	. . 3 |- ( ph -> ( x e. X |-> ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) ) : X --> RR )
19		ax-resscn 7988	. . . 4 |- RR C_ CC
20		ssid 3204	. . . . . . . . 9 |- CC C_ CC
21		cncfss 14903	. . . . . . . . 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 3182	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )
24	22, 5	sselid 3182	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )
25	23, 24	addcncf 14932	. . . . . 6 |- ( ph -> ( x e. X |-> ( A + B ) ) e. ( X -cn-> CC ) )
26		cncfss 14903	. . . . . . . . . 10 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( CC -cn-> RR ) C_ ( CC -cn-> CC ) )
27	19, 20, 26	mp2an 426	. . . . . . . . 9 |- ( CC -cn-> RR ) C_ ( CC -cn-> CC )
28		abscncf 14905	. . . . . . . . 9 |- abs e. ( CC -cn-> RR )
29	27, 28	sselii 3181	. . . . . . . 8 |- abs e. ( CC -cn-> CC )
30	29	a1i 9	. . . . . . 7 |- ( ph -> abs e. ( CC -cn-> CC ) )
31	23, 24	subcncf 14933	. . . . . . 7 |- ( ph -> ( x e. X |-> ( A - B ) ) e. ( X -cn-> CC ) )
32	30, 31	cncfmpt1f 14918	. . . . . 6 |- ( ph -> ( x e. X |-> ( abs ` ( A - B ) ) ) e. ( X -cn-> CC ) )
33	25, 32	subcncf 14933	. . . . 5 |- ( ph -> ( x e. X |-> ( ( A + B ) - ( abs ` ( A - B ) ) ) ) e. ( X -cn-> CC ) )
34		2cn 9078	. . . . . . 7 |- 2 e. CC
35		2ap0 9100	. . . . . . 7 |- 2 # 0
36		breq1 4037	. . . . . . . 8 |- ( y = 2 -> ( y # 0 <-> 2 # 0 ) )
37	36	elrab 2920	. . . . . . 7 |- ( 2 e. { y e. CC | y # 0 } <-> ( 2 e. CC /\ 2 # 0 ) )
38	34, 35, 37	mpbir2an 944	. . . . . 6 |- 2 e. { y e. CC | y # 0 }
39		cncfrss 14895	. . . . . . 7 |- ( ( x e. X |-> A ) e. ( X -cn-> RR ) -> X C_ CC )
40	1, 39	syl 14	. . . . . 6 |- ( ph -> X C_ CC )
41		apsscn 8691	. . . . . . 7 |- { y e. CC | y # 0 } C_ CC
42	41	a1i 9	. . . . . 6 |- ( ph -> { y e. CC | y # 0 } C_ CC )
43		cncfmptc 14916	. . . . . 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 1353	. . . . 5 |- ( ph -> ( x e. X |-> 2 ) e. ( X -cn-> { y e. CC | y # 0 } ) )
45	33, 44	divcncfap 14934	. . . 4 |- ( ph -> ( x e. X |-> ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) ) e. ( X -cn-> CC ) )
46		cncfcdm 14902	. . . 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 2273	1 |- ( ph -> ( x e. X |-> inf ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   {crab 2479   C_ wss 3157   {cpr 3624   class class class wbr 4034   |-> cmpt 4095   -->wf 5255   ` cfv 5259  (class class class)co 5925  infcinf 7058   CCcc 7894   RRcr 7895   0cc0 7896   + caddc 7899   < clt 8078   - cmin 8214   # cap 8625   / cdiv 8716   2c2 9058   abscabs 11179   -cn->ccncf 14890

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016  ax-addf 8018

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-map 6718  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-xneg 9864  df-xadd 9865  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-rest 12943  df-topgen 12962  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178  df-mopn 14179  df-top 14318  df-topon 14331  df-bases 14363  df-cn 14508  df-cnp 14509  df-tx 14573  df-cncf 14891

This theorem is referenced by:  hovercncf  14966