Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19453 | . . . 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 2823 | . . . 4 ⊢ (𝑋 · 𝑌) = (𝑋 · 𝑌) | |
8 | oveq1 7165 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦)) | |
9 | 8 | eqeq1d 2825 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑦) = (𝑋 · 𝑌))) |
10 | eleq1 2902 | . . . . . . 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 7166 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
14 | 13 | eqeq1d 2825 | . . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = (𝑋 · 𝑌) ↔ (𝑋 · 𝑌) = (𝑋 · 𝑌))) |
15 | eleq1 2902 | . . . . . . 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 3635 | . . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 Unitcui 19391 Irredcir 19392 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-irred 19395 |
This theorem is referenced by: prmirredlem 20642 |
Copyright terms: Public domain | W3C validator |