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Theorem isirred2 19443
Description: Expand out the class difference from isirred 19441. (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 3944 . . 3 (𝑋 ∈ (𝐵𝑈) ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈))
2 eldif 3944 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑈) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝑈))
3 eldif 3944 . . . . . . . . 9 (𝑦 ∈ (𝐵𝑈) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝑈))
42, 3anbi12i 628 . . . . . . . 8 ((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) ↔ ((𝑥𝐵 ∧ ¬ 𝑥𝑈) ∧ (𝑦𝐵 ∧ ¬ 𝑦𝑈)))
5 an4 654 . . . . . . . 8 (((𝑥𝐵 ∧ ¬ 𝑥𝑈) ∧ (𝑦𝐵 ∧ ¬ 𝑦𝑈)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)))
64, 5bitri 277 . . . . . . 7 ((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)))
76imbi1i 352 . . . . . 6 (((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
8 impexp 453 . . . . . . 7 ((((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋)))
9 pm4.56 985 . . . . . . . . . 10 ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) ↔ ¬ (𝑥𝑈𝑦𝑈))
10 df-ne 3015 . . . . . . . . . 10 ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
119, 10imbi12i 353 . . . . . . . . 9 (((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (¬ (𝑥𝑈𝑦𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
12 con34b 318 . . . . . . . . 9 (((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)) ↔ (¬ (𝑥𝑈𝑦𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
1311, 12bitr4i 280 . . . . . . . 8 (((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)))
1413imbi2i 338 . . . . . . 7 (((𝑥𝐵𝑦𝐵) → ((¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈) → (𝑥 · 𝑦) ≠ 𝑋)) ↔ ((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
158, 14bitri 277 . . . . . 6 ((((𝑥𝐵𝑦𝐵) ∧ (¬ 𝑥𝑈 ∧ ¬ 𝑦𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
167, 15bitri 277 . . . . 5 (((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
17162albii 1815 . . . 4 (∀𝑥𝑦((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
18 r2al 3199 . . . 4 (∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝑦((𝑥 ∈ (𝐵𝑈) ∧ 𝑦 ∈ (𝐵𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
19 r2al 3199 . . . 4 (∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)) ↔ ∀𝑥𝑦((𝑥𝐵𝑦𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
2017, 18, 193bitr4i 305 . . 3 (∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈)))
211, 20anbi12i 628 . 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 2819 . . 3 (𝐵𝑈) = (𝐵𝑈)
26 isirred2.4 . . 3 · = (.r𝑅)
2722, 23, 24, 25, 26isirred 19441 . 2 (𝑋𝐼 ↔ (𝑋 ∈ (𝐵𝑈) ∧ ∀𝑥 ∈ (𝐵𝑈)∀𝑦 ∈ (𝐵𝑈)(𝑥 · 𝑦) ≠ 𝑋))
28 df-3an 1084 . 2 ((𝑋𝐵 ∧ ¬ 𝑋𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))) ↔ ((𝑋𝐵 ∧ ¬ 𝑋𝑈) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
2921, 27, 283bitr4i 305 1 (𝑋𝐼 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1082  wal 1529   = wceq 1531  wcel 2108  wne 3014  wral 3136  cdif 3931  cfv 6348  (class class class)co 7148  Basecbs 16475  .rcmulr 16558  Unitcui 19381  Irredcir 19382
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-irred 19385
This theorem is referenced by:  irredcl  19446  irrednu  19447  irredmul  19451  prmirredlem  20632
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