Theorem isirred2 20372

Description: Expand out the class difference from isirred 20370. (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 3954	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
2		eldif 3954	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 𝑈) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈))
3		eldif 3954	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑈) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈))
4	2, 3	anbi12i 626	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈)))
5		an4 654	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)))
6	4, 5	bitri 274	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)))
7	6	imbi1i 348	. . . . . 6 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
8		impexp 449	. . . . . . 7 ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) → (𝑥 · 𝑦) ≠ 𝑋)))
9		pm4.56 986	. . . . . . . . . 10 ⊢ ((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) ↔ ¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))
10		df-ne 2930	. . . . . . . . . 10 ⊢ ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
11	9, 10	imbi12i 349	. . . . . . . . 9 ⊢ (((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
12		con34b 315	. . . . . . . . 9 ⊢ (((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ (¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
13	11, 12	bitr4i 277	. . . . . . . 8 ⊢ (((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
14	13	imbi2i 335	. . . . . . 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 274	. . . . . 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 274	. . . . 5 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
17	16	2albii 1814	. . . 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 3184	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥∀𝑦((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
19		r2al 3184	. . . 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 302	. . 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 626	. 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 2725	. . 3 ⊢ (𝐵 ∖ 𝑈) = (𝐵 ∖ 𝑈)
26		isirred2.4	. . 3 ⊢ · = (.r‘𝑅)
27	22, 23, 24, 25, 26	isirred 20370	. 2 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ (𝐵 ∖ 𝑈) ∧ ∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋))
28		df-3an 1086	. 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 302	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 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084  ∀wal 1531   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050   ∖ cdif 3941  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  ._rcmulr 17237  Unitcui 20306  Irredcir 20307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-irred 20310

This theorem is referenced by:  irredcl  20375  irrednu  20376  irredmul  20380  prmirredlem  21415  minplyirred  33509