Theorem refeq 14337

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 5358	. 2 |- ( ph -> F Fn RR )
3		refeq.g	. . 3 |- ( ph -> G : RR --> RR )
4	3	ffnd 5358	. 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 7933	. . . . . . . 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 5643	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( F ` x ) e. RR )
11	10	recnd 7960	. . . . . . . . . . . . 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 5643	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( G ` x ) e. RR )
14	13	recnd 7960	. . . . . . . . . . . . 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 8554	. . . . . . . . . . . 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 2366	. . . . . . . . 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 2563	. . . . . . . . . 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 663	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> -. 0 < x )
24	7, 8, 23	nltled 8052	. . . . . . 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 2563	. . . . . . . . . 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 663	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> -. x < 0 )
29	8, 7, 28	nltled 8052	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> 0 <_ x )
30	7, 8	letri3d 8047	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( x = 0 <-> ( x <_ 0 /\ 0 <_ x ) ) )
31	24, 29, 30	mpbir2and 944	. . . . . 6 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> x = 0 )
32	31	fveq2d 5511	. . . . 5 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) = ( F ` 0 ) )
33	31	fveq2d 5511	. . . . 5 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( G ` x ) = ( G ` 0 ) )
34	6, 32, 33	3eqtr4d 2218	. . . 4 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) = ( G ` x ) )
35	34, 19	pm2.65da 661	. . 3 |- ( ( ph /\ x e. RR ) -> -. ( F ` x ) # ( G ` x ) )
36		apti 8553	. . . 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 5608	1 |- ( ph -> F = G )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   =/= wne 2345   A.wral 2453   class class class wbr 3998   -->wf 5204   ` cfv 5208   CCcc 7784   RRcr 7785   0cc0 7786   < clt 7966   <_ cle 7967   # cap 8512

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513

This theorem is referenced by: (None)