Theorem isirred2 20324

Description: Expand out the class difference from isirred 20322. (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	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
isirred2	⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))

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

Step	Hyp	Ref	Expression
1		eldif 3915	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
2		eldif 3915	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 𝑈) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈))
3		eldif 3915	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑈) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈))
4	2, 3	anbi12i 628	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈)))
5		an4 656	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)))
6	4, 5	bitri 275	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)))
7	6	imbi1i 349	. . . . . 6 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
8		impexp 450	. . . . . . 7 ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) → (𝑥 · 𝑦) ≠ 𝑋)))
9		pm4.56 990	. . . . . . . . . 10 ⊢ ((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) ↔ ¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))
10		df-ne 2926	. . . . . . . . . 10 ⊢ ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
11	9, 10	imbi12i 350	. . . . . . . . 9 ⊢ (((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
12		con34b 316	. . . . . . . . 9 ⊢ (((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ (¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
13	11, 12	bitr4i 278	. . . . . . . 8 ⊢ (((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
14	13	imbi2i 336	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) → (𝑥 · 𝑦) ≠ 𝑋)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
15	8, 14	bitri 275	. . . . . 6 ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
16	7, 15	bitri 275	. . . . 5 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
17	16	2albii 1820	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
18		r2al 3165	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥∀𝑦((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
19		r2al 3165	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
20	17, 18, 19	3bitr4i 303	. . 3 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
21	1, 20	anbi12i 628	. 2 ⊢ ((𝑋 ∈ (𝐵 ∖ 𝑈) ∧ ∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
22		isirred2.1	. . 3 ⊢ 𝐵 = (Base‘𝑅)
23		isirred2.2	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
24		isirred2.3	. . 3 ⊢ 𝐼 = (Irred‘𝑅)
25		eqid 2729	. . 3 ⊢ (𝐵 ∖ 𝑈) = (𝐵 ∖ 𝑈)
26		isirred2.4	. . 3 ⊢ · = (.r‘𝑅)
27	22, 23, 24, 25, 26	isirred 20322	. 2 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ (𝐵 ∖ 𝑈) ∧ ∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋))
28		df-3an 1088	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))) ↔ ((𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
29	21, 27, 28	3bitr4i 303	1 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ∖ cdif 3902  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  ._rcmulr 17180  Unitcui 20258  Irredcir 20259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-irred 20262

This theorem is referenced by:  irredcl  20327  irrednu  20328  irredmul  20332  prmirredlem  21397  minplyirred  33680