Theorem ivthdich 15370

Description: The intermediate value theorem implies real number dichotomy. Because real number dichotomy (also known as analytic LLPO) is a constructive taboo, this means we will be unable to prove the intermediate value theorem as stated here (although versions with additional conditions, such as ivthinc 15360 for strictly monotonic functions, can be proved).

The proof is via a function which we call the hover function and which is also described in Section 5.1 of [Bauer], p. 493. Consider any real number z. We want to show that z <_ 0 \/ 0 <_ z. Because of hovercncf 15363, hovera 15364, and hoverb 15365, we are able to apply the intermediate value theorem to get a value c such that the hover function at c equals z. By axltwlin 8240, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 15366 or 0 <_ z by hovergt0 15367. (Contributed by Jim Kingdon and Mario Carneiro, 22-Jul-2025.)

Assertion

Ref	Expression
ivthdich	|- ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) -> A. r e. RR A. s e. RR ( r <_ s \/ s <_ r ) )

Distinct variable groups:   a, b, f, x   s, r

Proof of Theorem ivthdich

Dummy variables q t z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4090	. . . . . . . . . 10 |- ( x = q -> ( a < x <-> a < q ) )
2		breq1 4089	. . . . . . . . . 10 |- ( x = q -> ( x < b <-> q < b ) )
3		fveqeq2 5644	. . . . . . . . . 10 |- ( x = q -> ( ( f ` x ) = 0 <-> ( f ` q ) = 0 ) )
4	1, 2, 3	3anbi123d 1346	. . . . . . . . 9 |- ( x = q -> ( ( a < x /\ x < b /\ ( f ` x ) = 0 ) <-> ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) )
5	4	cbvrexv 2766	. . . . . . . 8 |- ( E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) <-> E. q e. RR ( a < q /\ q < b /\ ( f ` q ) = 0 ) )
6	5	imbi2i 226	. . . . . . 7 |- ( ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) <-> ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. q e. RR ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) )
7	6	2ralbii 2538	. . . . . 6 |- ( A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) <-> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. q e. RR ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) )
8	7	imbi2i 226	. . . . 5 |- ( ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) <-> ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. q e. RR ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) ) )
9	8	albii 1516	. . . 4 |- ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) <-> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. q e. RR ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) ) )
10		preq1 3746	. . . . . . . . 9 |- ( t = x -> { t , 0 } = { x , 0 } )
11	10	infeq1d 7205	. . . . . . . 8 |- ( t = x -> inf ( { t , 0 } , RR , < ) = inf ( { x , 0 } , RR , < ) )
12		oveq1 6020	. . . . . . . 8 |- ( t = x -> ( t - 1 ) = ( x - 1 ) )
13	11, 12	preq12d 3754	. . . . . . 7 |- ( t = x -> {inf ( { t , 0 } , RR , < ) , ( t - 1 ) } = {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } )
14	13	supeq1d 7180	. . . . . 6 |- ( t = x -> sup ( {inf ( { t , 0 } , RR , < ) , ( t - 1 ) } , RR , < ) = sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )
15	14	cbvmptv 4183	. . . . 5 |- ( t e. RR |-> sup ( {inf ( { t , 0 } , RR , < ) , ( t - 1 ) } , RR , < ) ) = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )
16		simpr 110	. . . . 5 |- ( ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. q e. RR ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) ) /\ z e. RR ) -> z e. RR )
17	9	biimpri 133	. . . . . 6 |- ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. q e. RR ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) ) -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )
18	17	adantr 276	. . . . 5 |- ( ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. q e. RR ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) ) /\ z e. RR ) -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )
19	15, 16, 18	ivthdichlem 15368	. . . 4 |- ( ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. q e. RR ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) ) /\ z e. RR ) -> ( z <_ 0 \/ 0 <_ z ) )
20	9, 19	sylanb 284	. . 3 |- ( ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) /\ z e. RR ) -> ( z <_ 0 \/ 0 <_ z ) )
21	20	ralrimiva 2603	. 2 |- ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) -> A. z e. RR ( z <_ 0 \/ 0 <_ z ) )
22		dich0 15369	. 2 |- ( A. z e. RR ( z <_ 0 \/ 0 <_ z ) <-> A. r e. RR A. s e. RR ( r <_ s \/ s <_ r ) )
23	21, 22	sylib 122	1 |- ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) -> A. r e. RR A. s e. RR ( r <_ s \/ s <_ r ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   A.wal 1393   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {cpr 3668   class class class wbr 4086   |-> cmpt 4148   ` cfv 5324  (class class class)co 6013   supcsup 7175  infcinf 7176   RRcr 8024   0cc0 8025   1c1 8026   < clt 8207   <_ cle 8208   - cmin 8343   -cn->ccncf 15287

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145  ax-addf 8147

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-map 6814  df-sup 7177  df-inf 7178  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-xneg 10000  df-xadd 10001  df-ioo 10120  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-rest 13317  df-topgen 13336  df-psmet 14550  df-xmet 14551  df-met 14552  df-bl 14553  df-mopn 14554  df-top 14715  df-topon 14728  df-bases 14760  df-cn 14905  df-cnp 14906  df-tx 14970  df-cncf 15288

This theorem is referenced by: (None)