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| Mirrors > Home > MPE Home > Th. List > irredmul | Structured version Visualization version GIF version | ||
| Description: If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| irredn0.i | ⊢ 𝐼 = (Irred‘𝑅) | 
| irredmul.b | ⊢ 𝐵 = (Base‘𝑅) | 
| irredmul.u | ⊢ 𝑈 = (Unit‘𝑅) | 
| irredmul.t | ⊢ · = (.r‘𝑅) | 
| Ref | Expression | 
|---|---|
| irredmul | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈)) | 
| 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 20421 | . . . 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 1148 | . . 3 ⊢ ((𝑋 · 𝑌) ∈ 𝐼 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))) | 
| 7 | eqid 2737 | . . . 4 ⊢ (𝑋 · 𝑌) = (𝑋 · 𝑌) | |
| 8 | oveq1 7438 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦)) | |
| 9 | 8 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑦) = (𝑋 · 𝑌))) | 
| 10 | eleq1 2829 | . . . . . . 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 344 | . . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑥 · 𝑦) = (𝑋 · 𝑌) → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)) ↔ ((𝑋 · 𝑦) = (𝑋 · 𝑌) → (𝑋 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))) | 
| 13 | oveq2 7439 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
| 14 | 13 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑌) = (𝑋 · 𝑌))) | 
| 15 | eleq1 2829 | . . . . . . 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 344 | . . . . 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 1118 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 .rcmulr 17298 Unitcui 20355 Irredcir 20356 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-irred 20359 | 
| This theorem is referenced by: prmirredlem 21483 mxidlirred 33500 irredminply 33757 | 
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