Theorem isirred2 20658

Description: Expand out the class difference from isirred 20656. (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 4048	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ 𝑈) ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈))
2		eldif 4048	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ 𝑈) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈))
3		eldif 4048	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∖ 𝑈) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈))
4	2, 3	anbi12i 640	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈)))
5		an4 669	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝑈) ∧ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)))
6	4, 5	bitri 278	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)))
7	6	imbi1i 352	. . . . . 6 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
8		impexp 456	. . . . . . 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 1004	. . . . . . . . . 10 ⊢ ((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) ↔ ¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))
10		df-ne 2956	. . . . . . . . . 10 ⊢ ((𝑥 · 𝑦) ≠ 𝑋 ↔ ¬ (𝑥 · 𝑦) = 𝑋)
11	9, 10	imbi12i 353	. . . . . . . . 9 ⊢ (((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ (¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
12		con34b 319	. . . . . . . . 9 ⊢ (((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ (¬ (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) → ¬ (𝑥 · 𝑦) = 𝑋))
13	11, 12	bitr4i 281	. . . . . . . 8 ⊢ (((¬ 𝑥 ∈ 𝑈 ∧ ¬ 𝑦 ∈ 𝑈) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
14	13	imbi2i 339	. . . . . . 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 278	. . . . . 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 278	. . . . 5 ⊢ (((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
17	16	2albii 1852	. . . 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 3198	. . . 4 ⊢ (∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋 ↔ ∀𝑥∀𝑦((𝑥 ∈ (𝐵 ∖ 𝑈) ∧ 𝑦 ∈ (𝐵 ∖ 𝑈)) → (𝑥 · 𝑦) ≠ 𝑋))
19		r2al 3198	. . . 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 306	. . 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 640	. 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 2760	. . 3 ⊢ (𝐵 ∖ 𝑈) = (𝐵 ∖ 𝑈)
26		isirred2.4	. . 3 ⊢ · = (.r‘𝑅)
27	22, 23, 24, 25, 26	isirred 20656	. 2 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ (𝐵 ∖ 𝑈) ∧ ∀𝑥 ∈ (𝐵 ∖ 𝑈)∀𝑦 ∈ (𝐵 ∖ 𝑈)(𝑥 · 𝑦) ≠ 𝑋))
28		df-3an 1105	. 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 306	1 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076   ∖ cdif 4044  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  ._rcmulr 17436  Unitcui 20592  Irredcir 20593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6496  df-fun 6543  df-fv 6549  df-ov 7425  df-irred 20596

This theorem is used by:  irredcl  20661  irrednu  20662  irredmul  20666  prmirredlem  21785  minplyirred  34391