Theorem ivthdich 15380

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 15370 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 15373, hovera 15374, and hoverb 15375, 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 8247, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 15376 or 0 <_ z by hovergt0 15377. (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 4092	. . . . . . . . . 10 |- ( x = q -> ( a < x <-> a < q ) )
2		breq1 4091	. . . . . . . . . 10 |- ( x = q -> ( x < b <-> q < b ) )
3		fveqeq2 5648	. . . . . . . . . 10 |- ( x = q -> ( ( f ` x ) = 0 <-> ( f ` q ) = 0 ) )
4	1, 2, 3	3anbi123d 1348	. . . . . . . . 9 |- ( x = q -> ( ( a < x /\ x < b /\ ( f ` x ) = 0 ) <-> ( a < q /\ q < b /\ ( f ` q ) = 0 ) ) )
5	4	cbvrexv 2768	. . . . . . . 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 2540	. . . . . 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 1518	. . . 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 3748	. . . . . . . . 9 |- ( t = x -> { t , 0 } = { x , 0 } )
11	10	infeq1d 7211	. . . . . . . 8 |- ( t = x -> inf ( { t , 0 } , RR , < ) = inf ( { x , 0 } , RR , < ) )
12		oveq1 6025	. . . . . . . 8 |- ( t = x -> ( t - 1 ) = ( x - 1 ) )
13	11, 12	preq12d 3756	. . . . . . 7 |- ( t = x -> {inf ( { t , 0 } , RR , < ) , ( t - 1 ) } = {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } )
14	13	supeq1d 7186	. . . . . 6 |- ( t = x -> sup ( {inf ( { t , 0 } , RR , < ) , ( t - 1 ) } , RR , < ) = sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )
15	14	cbvmptv 4185	. . . . 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 15378	. . . 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 2605	. 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 15379	. 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 715   /\ w3a 1004   A.wal 1395   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   {cpr 3670   class class class wbr 4088   |-> cmpt 4150   ` cfv 5326  (class class class)co 6018   supcsup 7181  infcinf 7182   RRcr 8031   0cc0 8032   1c1 8033   < clt 8214   <_ cle 8215   - cmin 8350   -cn->ccncf 15297

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152  ax-addf 8154

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-ioo 10127  df-seqfrec 10711  df-exp 10802  df-cj 11404  df-re 11405  df-im 11406  df-rsqrt 11560  df-abs 11561  df-rest 13326  df-topgen 13345  df-psmet 14560  df-xmet 14561  df-met 14562  df-bl 14563  df-mopn 14564  df-top 14725  df-topon 14738  df-bases 14770  df-cn 14915  df-cnp 14916  df-tx 14980  df-cncf 15298

This theorem is referenced by: (None)