Theorem refeq 14779

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 5367	. 2 ⊢ (𝜑 → 𝐹 Fn ℝ)
3		refeq.g	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
4	3	ffnd 5367	. 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 7958	. . . . . . . 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 5652	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
11	10	recnd 7986	. . . . . . . . . . . . 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 5652	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ)
14	13	recnd 7986	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℂ)
15	14	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐺‘𝑥) ∈ ℂ)
16		apne 8580	. . . . . . . . . . . 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 2368	. . . . . . . . 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 2565	. . . . . . . . . 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 663	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ¬ 0 < 𝑥)
24	7, 8, 23	nltled 8078	. . . . . . 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 2565	. . . . . . . . . 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 663	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → ¬ 𝑥 < 0)
29	8, 7, 28	nltled 8078	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 0 ≤ 𝑥)
30	7, 8	letri3d 8073	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝑥 = 0 ↔ (𝑥 ≤ 0 ∧ 0 ≤ 𝑥)))
31	24, 29, 30	mpbir2and 944	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 𝑥 = 0)
32	31	fveq2d 5520	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) = (𝐹‘0))
33	31	fveq2d 5520	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐺‘𝑥) = (𝐺‘0))
34	6, 32, 33	3eqtr4d 2220	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) = (𝐺‘𝑥))
35	34, 19	pm2.65da 661	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ¬ (𝐹‘𝑥) # (𝐺‘𝑥))
36		apti 8579	. . . 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 5617	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455   class class class wbr 4004  ⟶wf 5213  ‘cfv 5217  ℂcc 7809  ℝcr 7810  0cc0 7811   < clt 7992   ≤ cle 7993   # cap 8538

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539

This theorem is referenced by: (None)