Theorem ivthdich 15444

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 15434 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 15437, hovera 15438, and hoverb 15439, 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 8290, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 15440 or 0 <_ z by hovergt0 15441. (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 4097	. . . . . . . . . 10 |- ( x = q -> ( a < x <-> a < q ) )
2		breq1 4096	. . . . . . . . . 10 |- ( x = q -> ( x < b <-> q < b ) )
3		fveqeq2 5657	. . . . . . . . . 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 2769	. . . . . . . 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 2541	. . . . . 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 3752	. . . . . . . . 9 |- ( t = x -> { t , 0 } = { x , 0 } )
11	10	infeq1d 7254	. . . . . . . 8 |- ( t = x -> inf ( { t , 0 } , RR , < ) = inf ( { x , 0 } , RR , < ) )
12		oveq1 6035	. . . . . . . 8 |- ( t = x -> ( t - 1 ) = ( x - 1 ) )
13	11, 12	preq12d 3760	. . . . . . 7 |- ( t = x -> {inf ( { t , 0 } , RR , < ) , ( t - 1 ) } = {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } )
14	13	supeq1d 7229	. . . . . 6 |- ( t = x -> sup ( {inf ( { t , 0 } , RR , < ) , ( t - 1 ) } , RR , < ) = sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )
15	14	cbvmptv 4190	. . . . 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 15442	. . . 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 2606	. 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 15443	. 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 2202   A.wral 2511   E.wrex 2512   {cpr 3674   class class class wbr 4093   |-> cmpt 4155   ` cfv 5333  (class class class)co 6028   supcsup 7224  infcinf 7225   RRcr 8074   0cc0 8075   1c1 8076   < clt 8257   <_ cle 8258   - cmin 8393   -cn->ccncf 15361

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 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-n0 9446  df-z 9523  df-uz 9799  df-q 9897  df-rp 9932  df-xneg 10050  df-xadd 10051  df-ioo 10170  df-seqfrec 10754  df-exp 10845  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-rest 13385  df-topgen 13404  df-psmet 14619  df-xmet 14620  df-met 14621  df-bl 14622  df-mopn 14623  df-top 14789  df-topon 14802  df-bases 14834  df-cn 14979  df-cnp 14980  df-tx 15044  df-cncf 15362

This theorem is referenced by: (None)