Theorem dvdsrpropdg 14151

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
dvdsrpropdg.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
dvdsrpropdg.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
dvdsrpropdg.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
dvdsrpropdg.k	⊢ (𝜑 → 𝐾 ∈ SRing)
dvdsrpropdg.l	⊢ (𝜑 → 𝐿 ∈ SRing)

Assertion

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

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

Proof of Theorem dvdsrpropdg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsrpropdg.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
2	1	anassrs 400	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
3	2	eqeq1d 2238	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ (𝑥(.r‘𝐿)𝑦) = 𝑧))
4	3	an32s 568	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ (𝑥(.r‘𝐿)𝑦) = 𝑧))
5	4	rexbidva 2527	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧))
6	5	pm5.32da 452	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧)))
7		dvdsrpropdg.1	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
8	7	eleq2d 2299	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾)))
9	7	rexeqdv 2735	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧))
10	8, 9	anbi12d 473	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)))
11		dvdsrpropdg.2	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
12	11	eleq2d 2299	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐿)))
13	11	rexeqdv 2735	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧))
14	12, 13	anbi12d 473	. . . 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 218	. . 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 4153	. 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		eqidd 2230	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
18		eqidd 2230	. . 3 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐾))
19		dvdsrpropdg.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ SRing)
20		eqidd 2230	. . 3 ⊢ (𝜑 → (.r‘𝐾) = (.r‘𝐾))
21	17, 18, 19, 20	dvdsrvald 14097	. 2 ⊢ (𝜑 → (∥r‘𝐾) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)})
22		eqidd 2230	. . 3 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
23		eqidd 2230	. . 3 ⊢ (𝜑 → (∥r‘𝐿) = (∥r‘𝐿))
24		dvdsrpropdg.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ SRing)
25		eqidd 2230	. . 3 ⊢ (𝜑 → (.r‘𝐿) = (.r‘𝐿))
26	22, 23, 24, 25	dvdsrvald 14097	. 2 ⊢ (𝜑 → (∥r‘𝐿) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)})
27	16, 21, 26	3eqtr4d 2272	1 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  {copab 4147  ‘cfv 5324  (class class class)co 6013  Basecbs 13072  ._rcmulr 13151  SRingcsrg 13966  ∥_rcdsr 14089

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-mgp 13924  df-srg 13967  df-dvdsr 14092

This theorem is referenced by:  unitpropdg  14152