Theorem refeq 15831

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	⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
refeq.g	⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
refeq.lt0	⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑥 < 0 → (𝐹‘𝑥) = (𝐺‘𝑥)))
refeq.gt0	⊢ (𝜑 → ∀𝑥 ∈ ℝ (0 < 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥)))
refeq.0	⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))

Assertion

Ref	Expression
refeq	⊢ (𝜑 → 𝐹 = 𝐺)

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

Proof of Theorem refeq

Step	Hyp	Ref	Expression
1		refeq.f	. . 3 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
2	1	ffnd 5420	. 2 ⊢ (𝜑 → 𝐹 Fn ℝ)
3		refeq.g	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
4	3	ffnd 5420	. 2 ⊢ (𝜑 → 𝐺 Fn ℝ)
5		refeq.0	. . . . . 6 ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))
6	5	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘0) = (𝐺‘0))
7		simplr 528	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 𝑥 ∈ ℝ)
8		0red 8055	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 0 ∈ ℝ)
9		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) # (𝐺‘𝑥))
10	1	ffvelcdmda 5709	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
11	10	recnd 8083	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℂ)
12	11	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) ∈ ℂ)
13	3	ffvelcdmda 5709	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ)
14	13	recnd 8083	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℂ)
15	14	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐺‘𝑥) ∈ ℂ)
16		apne 8678	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ) → ((𝐹‘𝑥) # (𝐺‘𝑥) → (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
17	12, 15, 16	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ((𝐹‘𝑥) # (𝐺‘𝑥) → (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
18	9, 17	mpd 13	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) ≠ (𝐺‘𝑥))
19	18	neneqd 2396	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ¬ (𝐹‘𝑥) = (𝐺‘𝑥))
20		refeq.gt0	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (0 < 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥)))
21	20	r19.21bi 2593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (0 < 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥)))
22	21	adantr 276	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (0 < 𝑥 → (𝐹‘𝑥) = (𝐺‘𝑥)))
23	19, 22	mtod 664	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ¬ 0 < 𝑥)
24	7, 8, 23	nltled 8175	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 𝑥 ≤ 0)
25		refeq.lt0	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑥 < 0 → (𝐹‘𝑥) = (𝐺‘𝑥)))
26	25	r19.21bi 2593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥 < 0 → (𝐹‘𝑥) = (𝐺‘𝑥)))
27	26	adantr 276	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝑥 < 0 → (𝐹‘𝑥) = (𝐺‘𝑥)))
28	19, 27	mtod 664	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ¬ 𝑥 < 0)
29	8, 7, 28	nltled 8175	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 0 ≤ 𝑥)
30	7, 8	letri3d 8170	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝑥 = 0 ↔ (𝑥 ≤ 0 ∧ 0 ≤ 𝑥)))
31	24, 29, 30	mpbir2and 946	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 𝑥 = 0)
32	31	fveq2d 5574	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) = (𝐹‘0))
33	31	fveq2d 5574	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐺‘𝑥) = (𝐺‘0))
34	6, 32, 33	3eqtr4d 2247	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) = (𝐺‘𝑥))
35	34, 19	pm2.65da 662	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ¬ (𝐹‘𝑥) # (𝐺‘𝑥))
36		apti 8677	. . . 4 ⊢ (((𝐹‘𝑥) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ ¬ (𝐹‘𝑥) # (𝐺‘𝑥)))
37	11, 14, 36	syl2anc 411	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ ¬ (𝐹‘𝑥) # (𝐺‘𝑥)))
38	35, 37	mpbird 167	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) = (𝐺‘𝑥))
39	2, 4, 38	eqfnfvd 5674	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175   ≠ wne 2375  ∀wral 2483   class class class wbr 4043  ⟶wf 5264  ‘cfv 5268  ℂcc 7905  ℝcr 7906  0cc0 7907   < clt 8089   ≤ cle 8090   # cap 8636

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637

This theorem is referenced by: (None)