MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isirred2 Structured version   Visualization version   GIF version

Theorem isirred2 18472
Description: Expand out the class difference from isirred 18470. (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 3549 . . 3 (𝑋 ∈ (𝐵𝑈) ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
2 eldif 3549 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑈) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝑈))
3 eldif 3549 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑈) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝑈))
42, 3anbi12i 728 . . . . . . . 8 ((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) ↔ ((𝑥𝐵 ∧ ¬ 𝑥𝑈) ∧ (𝑦𝐵 ∧ ¬ 𝑦𝑈)))
5 an4 860 . . . . . . . 8 (((𝑥𝐵 ∧ ¬ 𝑥𝑈) ∧ (𝑦𝐵 ∧ ¬ 𝑦𝑈)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)))
64, 5bitri 262 . . . . . . 7 ((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)))
76imbi1i 337 . . . . . 6 (((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
8 impexp 460 . . . . . . 7 ((((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋)))
9 pm4.56 514 . . . . . . . . . 10 ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) ↔ ¬ (𝑥𝑈𝑦𝑈))
10 df-ne 2781 . . . . . . . . . 10 ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
119, 10imbi12i 338 . . . . . . . . 9 (((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (¬ (𝑥𝑈𝑦𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
12 con34b 304 . . . . . . . . 9 (((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)) ↔ (¬ (𝑥𝑈𝑦𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
1311, 12bitr4i 265 . . . . . . . 8 (((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)))
1413imbi2i 324 . . . . . . 7 (((𝑥𝐵𝑦𝐵) → ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋)) ↔ ((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
158, 14bitri 262 . . . . . 6 ((((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
167, 15bitri 262 . . . . 5 (((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
17162albii 1737 . . . 4 (∀𝑥𝑦((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
18 r2al 2922 . . . 4 (∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝑦((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
19 r2al 2922 . . . 4 (∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
2017, 18, 193bitr4i 290 . . 3 (∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)))
211, 20anbi12i 728 . 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 2609 . . 3 (𝐵𝑈) = (𝐵𝑈)
26 isirred2.4 . . 3 · = (.r𝑅)
2722, 23, 24, 25, 26isirred 18470 . 2 (𝑋𝐼 ↔ (𝑋 ∈ (𝐵𝑈) ∧ ∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋))
28 df-3an 1032 . 2 ((𝑋𝐵 ∧ ¬ 𝑋𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))) ↔ ((𝑋𝐵 ∧ ¬ 𝑋𝑈) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
2921, 27, 283bitr4i 290 1 (𝑋𝐼 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030  wal 1472   = wceq 1474  wcel 1976  wne 2779  wral 2895  cdif 3536  cfv 5789  (class class class)co 6526  Basecbs 15643  .rcmulr 15717  Unitcui 18410  Irredcir 18411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fv 5797  df-ov 6529  df-irred 18414
This theorem is referenced by:  irredcl  18475  irrednu  18476  irredmul  18480  prmirredlem  19607
  Copyright terms: Public domain W3C validator