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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 (𝜑𝐵 = (Base‘𝑅))
dvdsrvald.2 (𝜑 = (∥r𝑅))
dvdsrvald.r (𝜑𝑅 ∈ SRing)
dvdsrvald.3 (𝜑· = (.r𝑅))
Assertion
Ref Expression
dvdsrvald (𝜑 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (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.
StepHypRef Expression
1 df-dvdsr 13189 . . 3 r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦)})
2 fveq2 5514 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
32eleq2d 2247 . . . . 5 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
4 fveq2 5514 . . . . . . . 8 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
54oveqd 5889 . . . . . . 7 (𝑟 = 𝑅 → (𝑧(.r𝑟)𝑥) = (𝑧(.r𝑅)𝑥))
65eqeq1d 2186 . . . . . 6 (𝑟 = 𝑅 → ((𝑧(.r𝑟)𝑥) = 𝑦 ↔ (𝑧(.r𝑅)𝑥) = 𝑦))
72, 6rexeqbidv 2685 . . . . 5 (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦))
83, 7anbi12d 473 . . . 4 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)))
98opabbidv 4068 . . 3 (𝑟 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r𝑟)𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)})
10 dvdsrvald.r . . . 4 (𝜑𝑅 ∈ SRing)
1110elexd 2750 . . 3 (𝜑𝑅 ∈ V)
12 basfn 12512 . . . . . 6 Base Fn V
13 funfvex 5531 . . . . . . 7 ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
1413funfni 5315 . . . . . 6 ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
1512, 11, 14sylancr 414 . . . . 5 (𝜑 → (Base‘𝑅) ∈ V)
16 xpexg 4739 . . . . 5 (((Base‘𝑅) ∈ V ∧ (Base‘𝑅) ∈ V) → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
1715, 15, 16syl2anc 411 . . . 4 (𝜑 → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
18 simprr 531 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r𝑅)𝑥) = 𝑦)) → (𝑧(.r𝑅)𝑥) = 𝑦)
1910ad2antrr 488 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r𝑅)𝑥) = 𝑦)) → 𝑅 ∈ SRing)
20 simprl 529 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r𝑅)𝑥) = 𝑦)) → 𝑧 ∈ (Base‘𝑅))
21 simplr 528 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r𝑅)𝑥) = 𝑦)) → 𝑥 ∈ (Base‘𝑅))
22 eqid 2177 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
23 eqid 2177 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
2422, 23srgcl 13084 . . . . . . . . . 10 ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r𝑅)𝑥) ∈ (Base‘𝑅))
2519, 20, 21, 24syl3anc 1238 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r𝑅)𝑥) = 𝑦)) → (𝑧(.r𝑅)𝑥) ∈ (Base‘𝑅))
2618, 25eqeltrrd 2255 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r𝑅)𝑥) = 𝑦)) → 𝑦 ∈ (Base‘𝑅))
2726rexlimdvaa 2595 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦𝑦 ∈ (Base‘𝑅)))
2827imdistanda 448 . . . . . 6 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
2928ssopab2dv 4277 . . . . 5 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)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))}
3129, 30sseqtrrdi 3204 . . . 4 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
3217, 31ssexd 4142 . . 3 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)} ∈ V)
331, 9, 11, 32fvmptd3 5608 . 2 (𝜑 → (∥r𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)})
34 dvdsrvald.2 . 2 (𝜑 = (∥r𝑅))
35 dvdsrvald.1 . . . . 5 (𝜑𝐵 = (Base‘𝑅))
3635eleq2d 2247 . . . 4 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝑅)))
37 dvdsrvald.3 . . . . . . 7 (𝜑· = (.r𝑅))
3837oveqd 5889 . . . . . 6 (𝜑 → (𝑧 · 𝑥) = (𝑧(.r𝑅)𝑥))
3938eqeq1d 2186 . . . . 5 (𝜑 → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧(.r𝑅)𝑥) = 𝑦))
4035, 39rexeqbidv 2685 . . . 4 (𝜑 → (∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦))
4136, 40anbi12d 473 . . 3 (𝜑 → ((𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)))
4241opabbidv 4068 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r𝑅)𝑥) = 𝑦)})
4333, 34, 423eqtr4d 2220 1 (𝜑 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)})
Colors of variables: wff set class
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
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