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Theorem isirred2 18908
Description: Expand out the class difference from isirred 18906. (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 3731 . . 3 (𝑋 ∈ (𝐵𝑈) ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
2 eldif 3731 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑈) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝑈))
3 eldif 3731 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑈) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝑈))
42, 3anbi12i 604 . . . . . . . 8 ((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) ↔ ((𝑥𝐵 ∧ ¬ 𝑥𝑈) ∧ (𝑦𝐵 ∧ ¬ 𝑦𝑈)))
5 an4 627 . . . . . . . 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 437 . . . . . . 7 ((((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋)))
9 pm4.56 917 . . . . . . . . . 10 ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) ↔ ¬ (𝑥𝑈𝑦𝑈))
10 df-ne 2943 . . . . . . . . . 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 1895 . . . 4 (∀𝑥𝑦((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
18 r2al 3087 . . . 4 (∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝑦((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
19 r2al 3087 . . . 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 604 . 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 2770 . . 3 (𝐵𝑈) = (𝐵𝑈)
26 isirred2.4 . . 3 · = (.r𝑅)
2722, 23, 24, 25, 26isirred 18906 . 2 (𝑋𝐼 ↔ (𝑋 ∈ (𝐵𝑈) ∧ ∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋))
28 df-3an 1072 . 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  wa 382  wo 826  w3a 1070  wal 1628   = wceq 1630  wcel 2144  wne 2942  wral 3060  cdif 3718  cfv 6031  (class class class)co 6792  Basecbs 16063  .rcmulr 16149  Unitcui 18846  Irredcir 18847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-irred 18850
This theorem is referenced by:  irredcl  18911  irrednu  18912  irredmul  18916  prmirredlem  20055
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