Theorem irredmul 20046

Description: If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
irredn0.i	⊢ 𝐼 = (Irred‘𝑅)
irredmul.b	⊢ 𝐵 = (Base‘𝑅)
irredmul.u	⊢ 𝑈 = (Unit‘𝑅)
irredmul.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
irredmul	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

Proof of Theorem irredmul

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		irredmul.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
2		irredmul.u	. . . . 5 ⊢ 𝑈 = (Unit‘𝑅)
3		irredn0.i	. . . . 5 ⊢ 𝐼 = (Irred‘𝑅)
4		irredmul.t	. . . . 5 ⊢ · = (.r‘𝑅)
5	1, 2, 3, 4	isirred2 20038	. . . 4 ⊢ ((𝑋 · 𝑌) ∈ 𝐼 ↔ ((𝑋 · 𝑌) ∈ 𝐵 ∧ ¬ (𝑋 · 𝑌) ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
6	5	simp3bi 1146	. . 3 ⊢ ((𝑋 · 𝑌) ∈ 𝐼 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
7		eqid 2736	. . . 4 ⊢ (𝑋 · 𝑌) = (𝑋 · 𝑌)
8		oveq1 7344	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
9	8	eqeq1d 2738	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑦) = (𝑋 · 𝑌)))
10		eleq1 2824	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
11	10	orbi1d 914	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) ↔ (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
12	9, 11	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ ((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
13		oveq2 7345	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
14	13	eqeq1d 2738	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑌) = (𝑋 · 𝑌)))
15		eleq1 2824	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈))
16	15	orbi2d 913	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) ↔ (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈)))
17	14, 16	imbi12d 344	. . . . 5 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ ((𝑋 · 𝑌) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))))
18	12, 17	rspc2v 3579	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) → ((𝑋 · 𝑌) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))))
19	7, 18	mpii 46	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈)))
20	6, 19	syl5 34	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) ∈ 𝐼 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈)))
21	20	3impia 1116	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3061  ‘cfv 6479  (class class class)co 7337  Basecbs 17009  ._rcmulr 17060  Unitcui 19976  Irredcir 19977

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-ov 7340  df-irred 19980

This theorem is referenced by:  prmirredlem  20800