Theorem ivthdich 15510

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 15500 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 15503, hovera 15504, and hoverb 15505, 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 8340, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 15506 or 0 <_ z by hovergt0 15507. (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 4112	. . . . . . . . . 10 |- ( x = q -> ( a < x <-> a < q ) )
2		breq1 4111	. . . . . . . . . 10 |- ( x = q -> ( x < b <-> q < b ) )
3		fveqeq2 5678	. . . . . . . . . 10 |- ( x = q -> ( ( f ` x ) = 0 <-> ( f ` q ) = 0 ) )
4	1, 2, 3	3anbi123d 1349	. . . . . . . . 9 |- ( x = q -> ( ( a < x /\ x < b /\ ( f ` x ) = 0 ) <-> ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) )
5	4	cbvrexv 2778	. . . . . . . 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 2550	. . . . . 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 1519	. . . 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 3767	. . . . . . . . 9 |- ( t = x -> { t , 0 } = { x , 0 } )
11	10	infeq1d 7302	. . . . . . . 8 |- ( t = x -> inf ( { t , 0 } , RR , < ) = inf ( { x , 0 } , RR , < ) )
12		oveq1 6056	. . . . . . . 8 |- ( t = x -> ( t - 1 ) = ( x - 1 ) )
13	11, 12	preq12d 3775	. . . . . . 7 |- ( t = x -> {inf ( { t , 0 } , RR , < ) , ( t - 1 ) } = {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } )
14	13	supeq1d 7277	. . . . . 6 |- ( t = x -> sup ( {inf ( { t , 0 } , RR , < ) , ( t - 1 ) } , RR , < ) = sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )
15	14	cbvmptv 4205	. . . . 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 15508	. . . 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 2615	. 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 15509	. 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 716   /\ w3a 1005   A.wal 1396   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   {cpr 3689   class class class wbr 4108   |-> cmpt 4170   ` cfv 5351  (class class class)co 6049   supcsup 7272  infcinf 7273   RRcr 8125   0cc0 8126   1c1 8127   < clt 8307   <_ cle 8308   - cmin 8443   -cn->ccncf 15427

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246  ax-addf 8248

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-map 6883  df-sup 7274  df-inf 7275  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-xneg 10104  df-xadd 10105  df-ioo 10224  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-rest 13446  df-topgen 13465  df-psmet 14683  df-xmet 14684  df-met 14685  df-bl 14686  df-mopn 14687  df-top 14855  df-topon 14868  df-bases 14900  df-cn 15045  df-cnp 15046  df-tx 15110  df-cncf 15428

This theorem is referenced by: (None)