Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isidl Structured version   Visualization version   GIF version

Theorem isidl 37715
Description: The predicate "is an ideal of the ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idlval.1 𝐺 = (1st𝑅)
idlval.2 𝐻 = (2nd𝑅)
idlval.3 𝑋 = ran 𝐺
idlval.4 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
isidl (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑧,𝑋   𝑥,𝐼,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)   𝐻(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem isidl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 idlval.1 . . . 4 𝐺 = (1st𝑅)
2 idlval.2 . . . 4 𝐻 = (2nd𝑅)
3 idlval.3 . . . 4 𝑋 = ran 𝐺
4 idlval.4 . . . 4 𝑍 = (GId‘𝐺)
51, 2, 3, 4idlval 37714 . . 3 (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})
65eleq2d 2812 . 2 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ 𝐼 ∈ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))}))
71fvexi 6915 . . . . . . 7 𝐺 ∈ V
87rnex 7923 . . . . . 6 ran 𝐺 ∈ V
93, 8eqeltri 2822 . . . . 5 𝑋 ∈ V
109elpw2 5352 . . . 4 (𝐼 ∈ 𝒫 𝑋𝐼𝑋)
1110anbi1i 622 . . 3 ((𝐼 ∈ 𝒫 𝑋 ∧ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))) ↔ (𝐼𝑋 ∧ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
12 eleq2 2815 . . . . 5 (𝑖 = 𝐼 → (𝑍𝑖𝑍𝐼))
13 eleq2 2815 . . . . . . . 8 (𝑖 = 𝐼 → ((𝑥𝐺𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝐼))
1413raleqbi1dv 3323 . . . . . . 7 (𝑖 = 𝐼 → (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ↔ ∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼))
15 eleq2 2815 . . . . . . . . 9 (𝑖 = 𝐼 → ((𝑧𝐻𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝐼))
16 eleq2 2815 . . . . . . . . 9 (𝑖 = 𝐼 → ((𝑥𝐻𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝐼))
1715, 16anbi12d 630 . . . . . . . 8 (𝑖 = 𝐼 → (((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
1817ralbidv 3168 . . . . . . 7 (𝑖 = 𝐼 → (∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖) ↔ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
1914, 18anbi12d 630 . . . . . 6 (𝑖 = 𝐼 → ((∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)) ↔ (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
2019raleqbi1dv 3323 . . . . 5 (𝑖 = 𝐼 → (∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)) ↔ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
2112, 20anbi12d 630 . . . 4 (𝑖 = 𝐼 → ((𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))) ↔ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
2221elrab 3681 . . 3 (𝐼 ∈ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ↔ (𝐼 ∈ 𝒫 𝑋 ∧ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
23 3anass 1092 . . 3 ((𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))) ↔ (𝐼𝑋 ∧ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
2411, 22, 233bitr4i 302 . 2 (𝐼 ∈ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
256, 24bitrdi 286 1 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  {crab 3419  Vcvv 3462  wss 3947  𝒫 cpw 4607  ran crn 5683  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  GIdcgi 30423  RingOpscrngo 37595  Idlcidl 37708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-idl 37711
This theorem is referenced by:  isidlc  37716  idlss  37717  idl0cl  37719  idladdcl  37720  idllmulcl  37721  idlrmulcl  37722  rngoidl  37725  0idl  37726  intidl  37730  unichnidl  37732  keridl  37733
  Copyright terms: Public domain W3C validator