Theorem refeq 15523

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 5404	. 2 |- ( ph -> F Fn RR )
3		refeq.g	. . 3 |- ( ph -> G : RR --> RR )
4	3	ffnd 5404	. 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 8020	. . . . . . . 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 5693	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( F ` x ) e. RR )
11	10	recnd 8048	. . . . . . . . . . . . 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 5693	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( G ` x ) e. RR )
14	13	recnd 8048	. . . . . . . . . . . . 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 8642	. . . . . . . . . . . 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 2385	. . . . . . . . 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 2582	. . . . . . . . . 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 664	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> -. 0 < x )
24	7, 8, 23	nltled 8140	. . . . . . 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 2582	. . . . . . . . . 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 664	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> -. x < 0 )
29	8, 7, 28	nltled 8140	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> 0 <_ x )
30	7, 8	letri3d 8135	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( x = 0 <-> ( x <_ 0 /\ 0 <_ x ) ) )
31	24, 29, 30	mpbir2and 946	. . . . . 6 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> x = 0 )
32	31	fveq2d 5558	. . . . 5 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) = ( F ` 0 ) )
33	31	fveq2d 5558	. . . . 5 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( G ` x ) = ( G ` 0 ) )
34	6, 32, 33	3eqtr4d 2236	. . . 4 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) = ( G ` x ) )
35	34, 19	pm2.65da 662	. . 3 |- ( ( ph /\ x e. RR ) -> -. ( F ` x ) # ( G ` x ) )
36		apti 8641	. . . 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 5658	1 |- ( ph -> F = G )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   =/= wne 2364   A.wral 2472   class class class wbr 4029   -->wf 5250   ` cfv 5254   CCcc 7870   RRcr 7871   0cc0 7872   < clt 8054   <_ cle 8055   # cap 8600

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601

This theorem is referenced by: (None)