Theorem redc0 16687

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 8179	. . . . 5 ⊢ 0 ∈ ℝ
2		eqeq1 2238	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑧 = 𝑦))
3	2	dcbid 845	. . . . . 6 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑧 = 𝑦))
4		eqeq2 2241	. . . . . . 7 ⊢ (𝑦 = 0 → (𝑧 = 𝑦 ↔ 𝑧 = 0))
5	4	dcbid 845	. . . . . 6 ⊢ (𝑦 = 0 → (DECID 𝑧 = 𝑦 ↔ DECID 𝑧 = 0))
6	3, 5	rspc2v 2923	. . . . 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 2605	. 2 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ DECID 𝑥 = 𝑦 → ∀𝑧 ∈ ℝ DECID 𝑧 = 0)
10		eqeq1 2238	. . . . . 6 ⊢ (𝑧 = (𝑥 − 𝑦) → (𝑧 = 0 ↔ (𝑥 − 𝑦) = 0))
11	10	dcbid 845	. . . . 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 8443	. . . . . 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 2915	. . . 4 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → DECID (𝑥 − 𝑦) = 0)
16		simprl 531	. . . . . . 7 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
17	16	recnd 8208	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℂ)
18		simprr 533	. . . . . . 7 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
19	18	recnd 8208	. . . . . 6 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℂ)
20	17, 19	subeq0ad 8500	. . . . 5 ⊢ ((∀𝑧 ∈ ℝ DECID 𝑧 = 0 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → ((𝑥 − 𝑦) = 0 ↔ 𝑥 = 𝑦))
21	20	dcbid 845	. . . 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 2614	. 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 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510  (class class class)co 6018  ℝcr 8031  0cc0 8032   − cmin 8350

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-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-dc 842  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-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  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-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-sub 8352  df-neg 8353

This theorem is referenced by:  dcapnconstALT  16692