Theorem redc0 16336

Description: Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)

Assertion

Ref	Expression
redc0	⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ↔ ∀𝑧 ∈ ℝ DECID 𝑧 = 0)

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem redc0

Step	Hyp	Ref	Expression
1		0re 8114	. . . . 5 ⊢ 0 ∈ ℝ
2		eqeq1 2216	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
3	2	dcbid 842	. . . . . 6 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑧 = 𝑦))
4		eqeq2 2219	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑧 = 𝑦 ↔ 𝑧 = 0))
5	4	dcbid 842	. . . . . 6 ⊢ (𝑦 = 0 → (DECID 𝑧 = 𝑦 ↔ DECID 𝑧 = 0))
6	3, 5	rspc2v 2900	. . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 0 ∈ ℝ) → (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → DECID 𝑧 = 0))
7	1, 6	mpan2 425	. . . 4 ⊢ (𝑧 ∈ ℝ → (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → DECID 𝑧 = 0))
8	7	impcom 125	. . 3 ⊢ ((∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ∧ 𝑧 ∈ ℝ) → DECID 𝑧 = 0)
9	8	ralrimiva 2583	. 2 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
10		eqeq1 2216	. . . . . 6 ⊢ (𝑧 = (𝑥 − 𝑦) → (𝑧 = 0 ↔ (𝑥 − 𝑦) = 0))
11	10	dcbid 842	. . . . 5 ⊢ (𝑧 = (𝑥 − 𝑦) → (DECID 𝑧 = 0 ↔ DECID (𝑥 − 𝑦) = 0))
12		simpl 109	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
13		resubcl 8378	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ)
14	13	adantl 277	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 − 𝑦) ∈ ℝ)
15	11, 12, 14	rspcdva 2892	. . . 4 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → DECID (𝑥 − 𝑦) = 0)
16		simprl 529	. . . . . . 7 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
17	16	recnd 8143	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℂ)
18		simprr 531	. . . . . . 7 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
19	18	recnd 8143	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℂ)
20	17, 19	subeq0ad 8435	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥 − 𝑦) = 0 ↔ 𝑥 = 𝑦))
21	20	dcbid 842	. . . 4 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (DECID (𝑥 − 𝑦) = 0 ↔ DECID 𝑥 = 𝑦))
22	15, 21	mpbid 147	. . 3 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → DECID 𝑥 = 𝑦)
23	22	ralrimivva 2592	. 2 ⊢ (∀𝑧 ∈ ℝ DECID 𝑧 = 0 → ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦)
24	9, 23	impbii 126	1 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 ↔ ∀𝑧 ∈ ℝ DECID 𝑧 = 0)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 838   = wceq 1375   ∈ wcel 2180  ∀wral 2488  (class class class)co 5974  ℝcr 7966  0cc0 7967   − cmin 8285

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-setind 4606  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-iota 5254  df-fun 5296  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-sub 8287  df-neg 8288

This theorem is referenced by:  dcapnconstALT  16341