Theorem redc0 15617

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 8021	. . . . 5 ⊢ 0 ∈ ℝ
2		eqeq1 2200	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
3	2	dcbid 839	. . . . . 6 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑧 = 𝑦))
4		eqeq2 2203	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑧 = 𝑦 ↔ 𝑧 = 0))
5	4	dcbid 839	. . . . . 6 ⊢ (𝑦 = 0 → (DECID 𝑧 = 𝑦 ↔ DECID 𝑧 = 0))
6	3, 5	rspc2v 2878	. . . . 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 2567	. 2 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
10		eqeq1 2200	. . . . . 6 ⊢ (𝑧 = (𝑥 − 𝑦) → (𝑧 = 0 ↔ (𝑥 − 𝑦) = 0))
11	10	dcbid 839	. . . . 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 8285	. . . . . 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 2870	. . . 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 8050	. . . . . 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 8050	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℂ)
20	17, 19	subeq0ad 8342	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥 − 𝑦) = 0 ↔ 𝑥 = 𝑦))
21	20	dcbid 839	. . . 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 2576	. 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 835   = wceq 1364   ∈ wcel 2164  ∀wral 2472  (class class class)co 5919  ℝcr 7873  0cc0 7874   − cmin 8192

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-sub 8194  df-neg 8195

This theorem is referenced by:  dcapnconstALT  15622