Theorem refeq 16739

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 5490	. 2 ⊢ (𝜑 → 𝐹 Fn ℝ)
3		refeq.g	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
4	3	ffnd 5490	. 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 529	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 𝑥 ∈ ℝ)
8		0red 8223	. . . . . . . 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 5790	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
11	10	recnd 8250	. . . . . . . . . . . . 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 5790	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ)
14	13	recnd 8250	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℂ)
15	14	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐺‘𝑥) ∈ ℂ)
16		apne 8845	. . . . . . . . . . . 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 2424	. . . . . . . . 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 2621	. . . . . . . . . 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 669	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ¬ 0 < 𝑥)
24	7, 8, 23	nltled 8342	. . . . . . 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 2621	. . . . . . . . . 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 669	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ¬ 𝑥 < 0)
29	8, 7, 28	nltled 8342	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 0 ≤ 𝑥)
30	7, 8	letri3d 8337	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝑥 = 0 ↔ (𝑥 ≤ 0 ∧ 0 ≤ 𝑥)))
31	24, 29, 30	mpbir2and 953	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 𝑥 = 0)
32	31	fveq2d 5652	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) = (𝐹‘0))
33	31	fveq2d 5652	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐺‘𝑥) = (𝐺‘0))
34	6, 32, 33	3eqtr4d 2274	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) = (𝐺‘𝑥))
35	34, 19	pm2.65da 667	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ¬ (𝐹‘𝑥) # (𝐺‘𝑥))
36		apti 8844	. . . 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 5756	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202   ≠ wne 2403  ∀wral 2511   class class class wbr 4093  ⟶wf 5329  ‘cfv 5333  ℂcc 8073  ℝcr 8074  0cc0 8075   < clt 8256   ≤ cle 8257   # cap 8803

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804

This theorem is referenced by: (None)