Theorem isirred2 20357

Description: Expand out the class difference from isirred 20355. (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 3911	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
2		eldif 3911	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 𝑈) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈))
3		eldif 3911	. . . . . . . . 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 2933	. . . . . . . . . 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 1821	. . . 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 3172	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥∀𝑦((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
19		r2al 3172	. . . 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 2736	. . 3 ⊢ (𝐵 ∖ 𝑈) = (𝐵 ∖ 𝑈)
26		isirred2.4	. . 3 ⊢ · = (.r‘𝑅)
27	22, 23, 24, 25, 26	isirred 20355	. 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 1539   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051   ∖ cdif 3898  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  ._rcmulr 17178  Unitcui 20291  Irredcir 20292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-irred 20295

This theorem is referenced by:  irredcl  20360  irrednu  20361  irredmul  20365  prmirredlem  21427  minplyirred  33868