Theorem mincncf 15410

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 15371	. . . . . 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 5808	. . . 4 |- ( ( ph /\ x e. X ) -> A e. RR )
5		mincncf.b	. . . . . 6 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )
6		cncff 15371	. . . . . 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 5808	. . . 4 |- ( ( ph /\ x e. X ) -> B e. RR )
9		minabs 11859	. . . 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 4184	. 2 |- ( ph -> ( x e. X |-> inf ( { A , B } , RR , < ) ) = ( x e. X |-> ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) ) )
12	4, 8	readdcld 8251	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( A + B ) e. RR )
13	4, 8	resubcld 8602	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( A - B ) e. RR )
14	13	recnd 8250	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( A - B ) e. CC )
15	14	abscld 11804	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( abs ` ( A - B ) ) e. RR )
16	12, 15	resubcld 8602	. . . . 5 |- ( ( ph /\ x e. X ) -> ( ( A + B ) - ( abs ` ( A - B ) ) ) e. RR )
17	16	rehalfcld 9433	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) e. RR )
18	17	fmpttd 5810	. . 3 |- ( ph -> ( x e. X |-> ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) ) : X --> RR )
19		ax-resscn 8167	. . . 4 |- RR C_ CC
20		ssid 3248	. . . . . . . . 9 |- CC C_ CC
21		cncfss 15377	. . . . . . . . 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 3226	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )
24	22, 5	sselid 3226	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )
25	23, 24	addcncf 15406	. . . . . 6 |- ( ph -> ( x e. X |-> ( A + B ) ) e. ( X -cn-> CC ) )
26		cncfss 15377	. . . . . . . . . 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 15379	. . . . . . . . 9 |- abs e. ( CC -cn-> RR )
29	27, 28	sselii 3225	. . . . . . . 8 |- abs e. ( CC -cn-> CC )
30	29	a1i 9	. . . . . . 7 |- ( ph -> abs e. ( CC -cn-> CC ) )
31	23, 24	subcncf 15407	. . . . . . 7 |- ( ph -> ( x e. X |-> ( A - B ) ) e. ( X -cn-> CC ) )
32	30, 31	cncfmpt1f 15392	. . . . . 6 |- ( ph -> ( x e. X |-> ( abs ` ( A - B ) ) ) e. ( X -cn-> CC ) )
33	25, 32	subcncf 15407	. . . . 5 |- ( ph -> ( x e. X |-> ( ( A + B ) - ( abs ` ( A - B ) ) ) ) e. ( X -cn-> CC ) )
34		2cn 9256	. . . . . . 7 |- 2 e. CC
35		2ap0 9278	. . . . . . 7 |- 2 # 0
36		breq1 4096	. . . . . . . 8 |- ( y = 2 -> ( y # 0 <-> 2 # 0 ) )
37	36	elrab 2963	. . . . . . 7 |- ( 2 e. { y e. CC | y # 0 } <-> ( 2 e. CC /\ 2 # 0 ) )
38	34, 35, 37	mpbir2an 951	. . . . . 6 |- 2 e. { y e. CC | y # 0 }
39		cncfrss 15369	. . . . . . 7 |- ( ( x e. X |-> A ) e. ( X -cn-> RR ) -> X C_ CC )
40	1, 39	syl 14	. . . . . 6 |- ( ph -> X C_ CC )
41		apsscn 8869	. . . . . . 7 |- { y e. CC | y # 0 } C_ CC
42	41	a1i 9	. . . . . 6 |- ( ph -> { y e. CC | y # 0 } C_ CC )
43		cncfmptc 15390	. . . . . 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 1379	. . . . 5 |- ( ph -> ( x e. X |-> 2 ) e. ( X -cn-> { y e. CC | y # 0 } ) )
45	33, 44	divcncfap 15408	. . . 4 |- ( ph -> ( x e. X |-> ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) ) e. ( X -cn-> CC ) )
46		cncfcdm 15376	. . . 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 |-> inf ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   {crab 2515   C_ wss 3201   {cpr 3674   class class class wbr 4093   |-> cmpt 4155   -->wf 5329   ` cfv 5333  (class class class)co 6028  infcinf 7225   CCcc 8073   RRcr 8074   0cc0 8075   + caddc 8078   < clt 8256   - cmin 8392   # cap 8803   / cdiv 8894   2c2 9236   abscabs 11620   -cn->ccncf 15364

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195  ax-addf 8197

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-map 6862  df-sup 7226  df-inf 7227  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-xneg 10051  df-xadd 10052  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-rest 13387  df-topgen 13406  df-psmet 14622  df-xmet 14623  df-met 14624  df-bl 14625  df-mopn 14626  df-top 14792  df-topon 14805  df-bases 14837  df-cn 14982  df-cnp 14983  df-tx 15047  df-cncf 15365

This theorem is referenced by:  hovercncf  15440