Theorem refeq 17073

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 5534	. 2 ⊢ (𝜑 → 𝐹 Fn ℝ)
3		refeq.g	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
4	3	ffnd 5534	. 2 ⊢ (𝜑 → 𝐺 Fn ℝ)
5		refeq.0	. . . . . 6 ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))
6	5	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘0) = (𝐺‘0))
7		simplr 533	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 𝑥 ∈ ℝ)
8		0red 8327	. . . . . . . 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 5843	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
11	10	recnd 8354	. . . . . . . . . . . . 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 5843	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ)
14	13	recnd 8354	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℂ)
15	14	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐺‘𝑥) ∈ ℂ)
16		apne 8951	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ) → ((𝐹‘𝑥) # (𝐺‘𝑥) → (𝐹‘𝑥) ≠ (𝐺‘𝑥)))
17	12, 15, 16	syl2anc 415	. . . . . . . . . . 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 2441	. . . . . . . . 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 2638	. . . . . . . . . 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 673	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ¬ 0 < 𝑥)
24	7, 8, 23	nltled 8447	. . . . . . 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 2638	. . . . . . . . . 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 673	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ¬ 𝑥 < 0)
29	8, 7, 28	nltled 8447	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 0 ≤ 𝑥)
30	7, 8	letri3d 8441	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝑥 = 0 ↔ (𝑥 ≤ 0 ∧ 0 ≤ 𝑥)))
31	24, 29, 30	mpbir2and 957	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 𝑥 = 0)
32	31	fveq2d 5699	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) = (𝐹‘0))
33	31	fveq2d 5699	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐺‘𝑥) = (𝐺‘0))
34	6, 32, 33	3eqtr4d 2281	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) = (𝐺‘𝑥))
35	34, 19	pm2.65da 671	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ¬ (𝐹‘𝑥) # (𝐺‘𝑥))
36		apti 8950	. . . 4 ⊢ (((𝐹‘𝑥) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ ¬ (𝐹‘𝑥) # (𝐺‘𝑥)))
37	11, 14, 36	syl2anc 415	. . 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 5809	1 ⊢ (𝜑 → 𝐹 = 𝐺)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528   class class class wbr 4130  ⟶wf 5373  ‘cfv 5377  ℂcc 8177  ℝcr 8178  0cc0 8179   < clt 8360   ≤ cle 8361   # cap 8909

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910

This theorem is used by: (None)