Theorem isirred2 20369

Description: Expand out the class difference from isirred 20367. (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 3913	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
2		eldif 3913	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 𝑈) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈))
3		eldif 3913	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑈) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈))
4	2, 3	anbi12i 629	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈)))
5		an4 657	. . . . . . . 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		pm4.56 991	. . . . . . . . . 10 ⊢ ((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) ↔ ¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))
10		df-ne 2934	. . . . . . . . . 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 1822	. . . 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 3174	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥∀𝑦((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
19		r2al 3174	. . . 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 629	. 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 2737	. . 3 ⊢ (𝐵 ∖ 𝑈) = (𝐵 ∖ 𝑈)
26		isirred2.4	. . 3 ⊢ · = (.r‘𝑅)
27	22, 23, 24, 25, 26	isirred 20367	. 2 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ (𝐵 ∖ 𝑈) ∧ ∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋))
28		df-3an 1089	. 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 848   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∖ cdif 3900  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  ._rcmulr 17190  Unitcui 20303  Irredcir 20304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-irred 20307

This theorem is referenced by:  irredcl  20372  irrednu  20373  irredmul  20377  prmirredlem  21439  minplyirred  33888