Theorem dvdsrpropd 20697

Description: The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)

Hypotheses

Ref	Expression
rngidpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
rngidpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
rngidpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))

Assertion

Ref	Expression
dvdsrpropd	⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem dvdsrpropd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngidpropd.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
2	1	anassrs 473	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
3	2	eqeq1d 2765	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ (𝑥(.r‘𝐿)𝑦) = 𝑧))
4	3	an32s 665	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ (𝑥(.r‘𝐿)𝑦) = 𝑧))
5	4	rexbidva 3188	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧))
6	5	pm5.32da 590	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧)))
7		rngidpropd.1	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
8	7	eleq2d 2850	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾)))
9	7	rexeqdv 3324	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧))
10	8, 9	anbi12d 644	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)))
11		rngidpropd.2	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
12	11	eleq2d 2850	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐿)))
13	11	rexeqdv 3324	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧))
14	12, 13	anbi12d 644	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)))
15	6, 10, 14	3bitr3d 312	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)))
16	15	opabbidv 5177	. 2 ⊢ (𝜑 → {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)})
17		eqid 2763	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2763	. . 3 ⊢ (∥r‘𝐾) = (∥r‘𝐾)
19		eqid 2763	. . 3 ⊢ (.r‘𝐾) = (.r‘𝐾)
20	17, 18, 19	dvdsrval 20642	. 2 ⊢ (∥r‘𝐾) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)}
21		eqid 2763	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
22		eqid 2763	. . 3 ⊢ (∥r‘𝐿) = (∥r‘𝐿)
23		eqid 2763	. . 3 ⊢ (.r‘𝐿) = (.r‘𝐿)
24	21, 22, 23	dvdsrval 20642	. 2 ⊢ (∥r‘𝐿) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)}
25	16, 20, 24	3eqtr4g 2823	1 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {copab 5173  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  ._rcmulr 17473  ∥_rcdsr 20635

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-dvdsr 20638

This theorem is used by:  unitpropd  20698