Theorem dvdsrvald 14113

Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
dvdsrvald.1	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
dvdsrvald.2	⊢ (𝜑 → ∥ = (∥r‘𝑅))
dvdsrvald.r	⊢ (𝜑 → 𝑅 ∈ SRing)
dvdsrvald.3	⊢ (𝜑 → · = (.r‘𝑅))

Assertion

Ref	Expression
dvdsrvald	⊢ (𝜑 → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})

Distinct variable groups:   𝑥,𝑦, ∥   𝑥,𝑧,𝐵,𝑦   𝑥,𝑅,𝑦,𝑧   𝑥, · ,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hint:   ∥ (𝑧)

Proof of Theorem dvdsrvald

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvdsr 14108	. . 3 ⊢ ∥r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)})
2		fveq2 5639	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3	2	eleq2d 2301	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
4		fveq2 5639	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
5	4	oveqd 6035	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑧(.r‘𝑟)𝑥) = (𝑧(.r‘𝑅)𝑥))
6	5	eqeq1d 2240	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
7	2, 6	rexeqbidv 2747	. . . . 5 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
8	3, 7	anbi12d 473	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)))
9	8	opabbidv 4155	. . 3 ⊢ (𝑟 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
10		dvdsrvald.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ SRing)
11	10	elexd 2816	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
12		basfn 13146	. . . . . 6 ⊢ Base Fn V
13		funfvex 5656	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5432	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	12, 11, 14	sylancr 414	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
16		xpexg 4840	. . . . 5 ⊢ (((Base‘𝑅) ∈ V ∧ (Base‘𝑅) ∈ V) → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
17	15, 15, 16	syl2anc 411	. . . 4 ⊢ (𝜑 → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
18		simprr 533	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → (𝑧(.r‘𝑅)𝑥) = 𝑦)
19	10	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑅 ∈ SRing)
20		simprl 531	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑧 ∈ (Base‘𝑅))
21		simplr 529	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑥 ∈ (Base‘𝑅))
22		eqid 2231	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
23		eqid 2231	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
24	22, 23	srgcl 13989	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
25	19, 20, 21, 24	syl3anc 1273	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
26	18, 25	eqeltrrd 2309	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑦 ∈ (Base‘𝑅))
27	26	rexlimdvaa 2651	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦 → 𝑦 ∈ (Base‘𝑅)))
28	27	imdistanda 448	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
29	28	ssopab2dv 4373	. . . . 5 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))})
30		df-xp 4731	. . . . 5 ⊢ ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
31	29, 30	sseqtrrdi 3276	. . . 4 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
32	17, 31	ssexd 4229	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ∈ V)
33	1, 9, 11, 32	fvmptd3 5740	. 2 ⊢ (𝜑 → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
34		dvdsrvald.2	. 2 ⊢ (𝜑 → ∥ = (∥r‘𝑅))
35		dvdsrvald.1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
36	35	eleq2d 2301	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝑅)))
37		dvdsrvald.3	. . . . . . 7 ⊢ (𝜑 → · = (.r‘𝑅))
38	37	oveqd 6035	. . . . . 6 ⊢ (𝜑 → (𝑧 · 𝑥) = (𝑧(.r‘𝑅)𝑥))
39	38	eqeq1d 2240	. . . . 5 ⊢ (𝜑 → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
40	35, 39	rexeqbidv 2747	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
41	36, 40	anbi12d 473	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)))
42	41	opabbidv 4155	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
43	33, 34, 42	3eqtr4d 2274	1 ⊢ (𝜑 → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∃wrex 2511  Vcvv 2802  {copab 4149   × cxp 4723   Fn wfn 5321  ‘cfv 5326  (class class class)co 6018  Basecbs 13087  ._rcmulr 13166  SRingcsrg 13982  ∥_rcdsr 14105

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  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-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  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-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13090  df-slot 13091  df-base 13093  df-sets 13094  df-plusg 13178  df-mulr 13179  df-0g 13346  df-mgm 13444  df-sgrp 13490  df-mnd 13505  df-mgp 13940  df-srg 13983  df-dvdsr 14108

This theorem is referenced by:  dvdsrd  14114  dvdsrex  14118  dvdsrpropdg  14167  dvdsrzring  14623