Theorem refeq 16004

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 5428	. 2 |- ( ph -> F Fn RR )
3		refeq.g	. . 3 |- ( ph -> G : RR --> RR )
4	3	ffnd 5428	. 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 8075	. . . . . . . 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 5717	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( F ` x ) e. RR )
11	10	recnd 8103	. . . . . . . . . . . . 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 5717	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR ) -> ( G ` x ) e. RR )
14	13	recnd 8103	. . . . . . . . . . . . 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 8698	. . . . . . . . . . . 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 2397	. . . . . . . . 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 2594	. . . . . . . . . 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 665	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> -. 0 < x )
24	7, 8, 23	nltled 8195	. . . . . . 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 2594	. . . . . . . . . 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 665	. . . . . . . 8 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> -. x < 0 )
29	8, 7, 28	nltled 8195	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> 0 <_ x )
30	7, 8	letri3d 8190	. . . . . . 7 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( x = 0 <-> ( x <_ 0 /\ 0 <_ x ) ) )
31	24, 29, 30	mpbir2and 947	. . . . . 6 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> x = 0 )
32	31	fveq2d 5582	. . . . 5 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) = ( F ` 0 ) )
33	31	fveq2d 5582	. . . . 5 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( G ` x ) = ( G ` 0 ) )
34	6, 32, 33	3eqtr4d 2248	. . . 4 |- ( ( ( ph /\ x e. RR ) /\ ( F ` x ) # ( G ` x ) ) -> ( F ` x ) = ( G ` x ) )
35	34, 19	pm2.65da 663	. . 3 |- ( ( ph /\ x e. RR ) -> -. ( F ` x ) # ( G ` x ) )
36		apti 8697	. . . 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 5682	1 |- ( ph -> F = G )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   =/= wne 2376   A.wral 2484   class class class wbr 4045   -->wf 5268   ` cfv 5272   CCcc 7925   RRcr 7926   0cc0 7927   < clt 8109   <_ cle 8110   # cap 8656

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657

This theorem is referenced by: (None)