Theorem refeq 16396

Description: Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.)

Hypotheses

Ref	Expression
refeq.f	|- ( ph -> F : RR --> RR )
refeq.g	|- ( ph -> G : RR --> RR )
refeq.lt0	|- ( ph -> A. x e. RR ( x < 0 -> ( F ` x ) = ( G ` x ) ) )
refeq.gt0	|- ( ph -> A. x e. RR ( 0 < x -> ( F ` x ) = ( G ` x ) ) )
refeq.0	|- ( ph -> ( F ` 0 ) = ( G ` 0 ) )

Assertion

Ref	Expression
refeq	|- ( ph -> F = G )

Distinct variable groups:   x, F   x, G   ph, x

Proof of Theorem refeq

Step	Hyp	Ref	Expression
1		refeq.f	. . 3 |- ( ph -> F : RR --> RR )
2	1	ffnd 5474	. 2 |- ( ph -> F Fn RR )
3		refeq.g	. . 3 |- ( ph -> G : RR --> RR )
4	3	ffnd 5474	. 2 |- ( ph -> G Fn RR )
5		refeq.0	. . . . . 6 |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )
6	5	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` 0 ) = ( G ` 0 ) )
7		simplr 528	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> x e. RR )
8		0red 8147	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> 0 e. RR )
9		simpr 110	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) # ( G ` x ) )
10	1	ffvelcdmda 5770	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( F ` x ) e. RR )
11	10	recnd 8175	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. RR ) -> ( F ` x ) e. CC )
12	11	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) e. CC )
13	3	ffvelcdmda 5770	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( G ` x ) e. RR )
14	13	recnd 8175	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. RR ) -> ( G ` x ) e. CC )
15	14	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( G ` x ) e. CC )
16		apne 8770	. . . . . . . . . . . 12 |- ( ( ( F ` x ) e. CC /\ ( G ` x ) e. CC ) -> ( ( F ` x ) # ( G ` x ) -> ( F ` x ) =/= ( G ` x ) ) )
17	12, 15, 16	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( ( F ` x ) # ( G ` x ) -> ( F ` x ) =/= ( G ` x ) ) )
18	9, 17	mpd 13	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) =/= ( G ` x ) )
19	18	neneqd 2421	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> -. ( F ` x ) = ( G ` x ) )
20		refeq.gt0	. . . . . . . . . . 11 |- ( ph -> A. x e. RR ( 0 < x -> ( F ` x ) = ( G ` x ) ) )
21	20	r19.21bi 2618	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> ( 0 < x -> ( F ` x ) = ( G ` x ) ) )
22	21	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( 0 < x -> ( F ` x ) = ( G ` x ) ) )
23	19, 22	mtod 667	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> -. 0 < x )
24	7, 8, 23	nltled 8267	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> x <_ 0 )
25		refeq.lt0	. . . . . . . . . . 11 |- ( ph -> A. x e. RR ( x < 0 -> ( F ` x ) = ( G ` x ) ) )
26	25	r19.21bi 2618	. . . . . . . . . 10 |- ( ( ph /\ x e. RR ) -> ( x < 0 -> ( F ` x ) = ( G ` x ) ) )
27	26	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( x < 0 -> ( F ` x ) = ( G ` x ) ) )
28	19, 27	mtod 667	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> -. x < 0 )
29	8, 7, 28	nltled 8267	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> 0 <_ x )
30	7, 8	letri3d 8262	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( x = 0 <-> ( x <_ 0 /\ 0 <_ x ) ) )
31	24, 29, 30	mpbir2and 950	. . . . . 6 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> x = 0 )
32	31	fveq2d 5631	. . . . 5 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) = ( F ` 0 ) )
33	31	fveq2d 5631	. . . . 5 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( G ` x ) = ( G ` 0 ) )
34	6, 32, 33	3eqtr4d 2272	. . . 4 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) = ( G ` x ) )
35	34, 19	pm2.65da 665	. . 3 |- ( ( ph /\ x e. RR ) -> -. ( F ` x ) # ( G ` x ) )
36		apti 8769	. . . 4 |- ( ( ( F ` x ) e. CC /\ ( G ` x ) e. CC ) -> ( ( F ` x ) = ( G ` x ) <-> -. ( F ` x ) # ( G ` x ) ) )
37	11, 14, 36	syl2anc 411	. . 3 |- ( ( ph /\ x e. RR ) -> ( ( F ` x ) = ( G ` x ) <-> -. ( F ` x ) # ( G ` x ) ) )
38	35, 37	mpbird 167	. 2 |- ( ( ph /\ x e. RR ) -> ( F ` x ) = ( G ` x ) )
39	2, 4, 38	eqfnfvd 5735	1 |- ( ph -> F = G )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   class class class wbr 4083   -->wf 5314   ` cfv 5318   CCcc 7997   RRcr 7998   0cc0 7999   < clt 8181   <_ cle 8182   # cap 8728

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-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116

This theorem depends on definitions:  df-bi 117  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-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729

This theorem is referenced by: (None)