Theorem irredmul 19681

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 19673	. . . 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 1149	. . 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 7198	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
9	8	eqeq1d 2738	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑦) = (𝑋 · 𝑌)))
10		eleq1 2818	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
11	10	orbi1d 917	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) ↔ (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
12	9, 11	imbi12d 348	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ ((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
13		oveq2 7199	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
14	13	eqeq1d 2738	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑌) = (𝑋 · 𝑌)))
15		eleq1 2818	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈))
16	15	orbi2d 916	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) ↔ (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈)))
17	14, 16	imbi12d 348	. . . . 5 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ ((𝑋 · 𝑌) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))))
18	12, 17	rspc2v 3537	. . . 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 1119	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  ._rcmulr 16750  Unitcui 19611  Irredcir 19612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6316  df-fun 6360  df-fv 6366  df-ov 7194  df-irred 19615

This theorem is referenced by:  prmirredlem  20413