Theorem irredmul 19459

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 19451	. . . 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 1143	. . 3 ⊢ ((𝑋 · 𝑌) ∈ 𝐼 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
7		eqid 2821	. . . 4 ⊢ (𝑋 · 𝑌) = (𝑋 · 𝑌)
8		oveq1 7163	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
9	8	eqeq1d 2823	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑦) = (𝑋 · 𝑌)))
10		eleq1 2900	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
11	10	orbi1d 913	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) ↔ (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))
12	9, 11	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ ((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
13		oveq2 7164	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
14	13	eqeq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑌) = (𝑋 · 𝑌)))
15		eleq1 2900	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈))
16	15	orbi2d 912	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈) ↔ (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈)))
17	14, 16	imbi12d 347	. . . . 5 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ ((𝑋 · 𝑌) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))))
18	12, 17	rspc2v 3633	. . . 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 1113	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  ._rcmulr 16566  Unitcui 19389  Irredcir 19390

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-irred 19393

This theorem is referenced by:  prmirredlem  20640