Theorem refeq 16632

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 5483	. 2 ⊢ (𝜑 → 𝐹 Fn ℝ)
3		refeq.g	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
4	3	ffnd 5483	. 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 8179	. . . . . . . 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 5782	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
11	10	recnd 8207	. . . . . . . . . . . . 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 5782	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℝ)
14	13	recnd 8207	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) ∈ ℂ)
15	14	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐺‘𝑥) ∈ ℂ)
16		apne 8802	. . . . . . . . . . . 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 2423	. . . . . . . . 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 2620	. . . . . . . . . 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 8299	. . . . . . 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 2620	. . . . . . . . . 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 8299	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 0 ≤ 𝑥)
30	7, 8	letri3d 8294	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝑥 = 0 ↔ (𝑥 ≤ 0 ∧ 0 ≤ 𝑥)))
31	24, 29, 30	mpbir2and 952	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → 𝑥 = 0)
32	31	fveq2d 5643	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ (𝐹‘𝑥) # (𝐺‘𝑥)) → (𝐹‘𝑥) = (𝐹‘0))
33	31	fveq2d 5643	. . . . 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 8801	. . . 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 5747	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  ∀wral 2510   class class class wbr 4088  ⟶wf 5322  ‘cfv 5326  ℂcc 8029  ℝcr 8030  0cc0 8031   < clt 8213   ≤ cle 8214   # cap 8760

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761

This theorem is referenced by: (None)