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Theorem isirred2 18747
Description: Expand out the class difference from isirred 18745. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
isirred2.1 𝐵 = (Base‘𝑅)
isirred2.2 𝑈 = (Unit‘𝑅)
isirred2.3 𝐼 = (Irred‘𝑅)
isirred2.4 · = (.r𝑅)
Assertion
Ref Expression
isirred2 (𝑋𝐼 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   · (𝑥,𝑦)   𝐼(𝑥,𝑦)

Proof of Theorem isirred2
StepHypRef Expression
1 eldif 3617 . . 3 (𝑋 ∈ (𝐵𝑈) ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
2 eldif 3617 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑈) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝑈))
3 eldif 3617 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑈) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝑈))
42, 3anbi12i 733 . . . . . . . 8 ((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) ↔ ((𝑥𝐵 ∧ ¬ 𝑥𝑈) ∧ (𝑦𝐵 ∧ ¬ 𝑦𝑈)))
5 an4 882 . . . . . . . 8 (((𝑥𝐵 ∧ ¬ 𝑥𝑈) ∧ (𝑦𝐵 ∧ ¬ 𝑦𝑈)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)))
64, 5bitri 264 . . . . . . 7 ((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)))
76imbi1i 338 . . . . . 6 (((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
8 impexp 461 . . . . . . 7 ((((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋)))
9 pm4.56 515 . . . . . . . . . 10 ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) ↔ ¬ (𝑥𝑈𝑦𝑈))
10 df-ne 2824 . . . . . . . . . 10 ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
119, 10imbi12i 339 . . . . . . . . 9 (((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (¬ (𝑥𝑈𝑦𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
12 con34b 305 . . . . . . . . 9 (((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)) ↔ (¬ (𝑥𝑈𝑦𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
1311, 12bitr4i 267 . . . . . . . 8 (((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)))
1413imbi2i 325 . . . . . . 7 (((𝑥𝐵𝑦𝐵) → ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋)) ↔ ((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
158, 14bitri 264 . . . . . 6 ((((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
167, 15bitri 264 . . . . 5 (((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
17162albii 1788 . . . 4 (∀𝑥𝑦((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
18 r2al 2968 . . . 4 (∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝑦((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
19 r2al 2968 . . . 4 (∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
2017, 18, 193bitr4i 292 . . 3 (∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)))
211, 20anbi12i 733 . 2 ((𝑋 ∈ (𝐵𝑈) ∧ ∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑋𝐵 ∧ ¬ 𝑋𝑈) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
22 isirred2.1 . . 3 𝐵 = (Base‘𝑅)
23 isirred2.2 . . 3 𝑈 = (Unit‘𝑅)
24 isirred2.3 . . 3 𝐼 = (Irred‘𝑅)
25 eqid 2651 . . 3 (𝐵𝑈) = (𝐵𝑈)
26 isirred2.4 . . 3 · = (.r𝑅)
2722, 23, 24, 25, 26isirred 18745 . 2 (𝑋𝐼 ↔ (𝑋 ∈ (𝐵𝑈) ∧ ∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋))
28 df-3an 1056 . 2 ((𝑋𝐵 ∧ ¬ 𝑋𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))) ↔ ((𝑋𝐵 ∧ ¬ 𝑋𝑈) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
2921, 27, 283bitr4i 292 1 (𝑋𝐼 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wne 2823  wral 2941  cdif 3604  cfv 5926  (class class class)co 6690  Basecbs 15904  .rcmulr 15989  Unitcui 18685  Irredcir 18686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-irred 18689
This theorem is referenced by:  irredcl  18750  irrednu  18751  irredmul  18755  prmirredlem  19889
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