Theorem dvdsrvald 13193

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 13189	. . 3 ⊢ ∥r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)})
2		fveq2 5514	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3	2	eleq2d 2247	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
4		fveq2 5514	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
5	4	oveqd 5889	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑧(.r‘𝑟)𝑥) = (𝑧(.r‘𝑅)𝑥))
6	5	eqeq1d 2186	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
7	2, 6	rexeqbidv 2685	. . . . 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 4068	. . 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 2750	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
12		basfn 12512	. . . . . 6 ⊢ Base Fn V
13		funfvex 5531	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5315	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	12, 11, 14	sylancr 414	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
16		xpexg 4739	. . . . 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 531	. . . . . . . . 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 529	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑧 ∈ (Base‘𝑅))
21		simplr 528	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑥 ∈ (Base‘𝑅))
22		eqid 2177	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
23		eqid 2177	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
24	22, 23	srgcl 13084	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
25	19, 20, 21, 24	syl3anc 1238	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
26	18, 25	eqeltrrd 2255	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑦 ∈ (Base‘𝑅))
27	26	rexlimdvaa 2595	. . . . . . 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 4277	. . . . 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 4631	. . . . 5 ⊢ ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
31	29, 30	sseqtrrdi 3204	. . . 4 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
32	17, 31	ssexd 4142	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ∈ V)
33	1, 9, 11, 32	fvmptd3 5608	. 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 2247	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝑅)))
37		dvdsrvald.3	. . . . . . 7 ⊢ (𝜑 → · = (.r‘𝑅))
38	37	oveqd 5889	. . . . . 6 ⊢ (𝜑 → (𝑧 · 𝑥) = (𝑧(.r‘𝑅)𝑥))
39	38	eqeq1d 2186	. . . . 5 ⊢ (𝜑 → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
40	35, 39	rexeqbidv 2685	. . . 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 4068	. 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 2220	1 ⊢ (𝜑 → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  ∃wrex 2456  Vcvv 2737  {copab 4062   × cxp 4623   Fn wfn 5210  ‘cfv 5215  (class class class)co 5872  Basecbs 12454  ._rcmulr 12529  SRingcsrg 13077  ∥_rcdsr 13186

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-mgp 13062  df-srg 13078  df-dvdsr 13189

This theorem is referenced by:  dvdsrd  13194  dvdsrex  13198  dvdsrpropdg  13247  dvdsrzring  13362