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Metamath Proof Explorer

Theorem uvcf1

Description: In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015) (Revised by Mario Carneiro, 14-Jun-2015)

Ref Expression
Hypotheses uvcff.u ⊢ 𝑈 = ( 𝑅 unitVec 𝐼 )
uvcff.y ⊢ 𝑌 = ( 𝑅 freeLMod 𝐼 )
uvcff.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
Assertion uvcf1 ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) → 𝑈 : 𝐼 –1-1→ 𝐵 )

Proof

Step Hyp Ref Expression
1 uvcff.u ⊢ 𝑈 = ( 𝑅 unitVec 𝐼 )
2 uvcff.y ⊢ 𝑌 = ( 𝑅 freeLMod 𝐼 )
3 uvcff.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
4 nzrring ⊢ ( 𝑅 ∈ NzRing → 𝑅 ∈ Ring )
5 1 2 3 uvcff ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ) → 𝑈 : 𝐼 ⟶ 𝐵 )
6 4 5 sylan ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) → 𝑈 : 𝐼 ⟶ 𝐵 )
7 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
8 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
9 7 8 nzrnz ⊢ ( 𝑅 ∈ NzRing → ( 1r ‘ 𝑅 ) ≠ ( 0g ‘ 𝑅 ) )
10 9 ad3antrrr ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → ( 1r ‘ 𝑅 ) ≠ ( 0g ‘ 𝑅 ) )
11 4 ad3antrrr ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → 𝑅 ∈ Ring )
12 simpllr ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → 𝐼 ∈ 𝑊 )
13 simplrl ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → 𝑖 ∈ 𝐼 )
14 1 11 12 13 7 uvcvv1 ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → ( ( 𝑈 ‘ 𝑖 ) ‘ 𝑖 ) = ( 1r ‘ 𝑅 ) )
15 simplrr ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → 𝑗 ∈ 𝐼 )
16 simpr ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → 𝑖 ≠ 𝑗 )
17 16 necomd ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → 𝑗 ≠ 𝑖 )
18 1 11 12 15 13 17 8 uvcvv0 ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → ( ( 𝑈 ‘ 𝑗 ) ‘ 𝑖 ) = ( 0g ‘ 𝑅 ) )
19 10 14 18 3netr4d ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → ( ( 𝑈 ‘ 𝑖 ) ‘ 𝑖 ) ≠ ( ( 𝑈 ‘ 𝑗 ) ‘ 𝑖 ) )
20 fveq1 ⊢ ( ( 𝑈 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑗 ) → ( ( 𝑈 ‘ 𝑖 ) ‘ 𝑖 ) = ( ( 𝑈 ‘ 𝑗 ) ‘ 𝑖 ) )
21 20 necon3i ⊢ ( ( ( 𝑈 ‘ 𝑖 ) ‘ 𝑖 ) ≠ ( ( 𝑈 ‘ 𝑗 ) ‘ 𝑖 ) → ( 𝑈 ‘ 𝑖 ) ≠ ( 𝑈 ‘ 𝑗 ) )
22 19 21 syl ⊢ ( ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) ∧ 𝑖 ≠ 𝑗 ) → ( 𝑈 ‘ 𝑖 ) ≠ ( 𝑈 ‘ 𝑗 ) )
23 22 ex ⊢ ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) → ( 𝑖 ≠ 𝑗 → ( 𝑈 ‘ 𝑖 ) ≠ ( 𝑈 ‘ 𝑗 ) ) )
24 23 necon4d ⊢ ( ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐼 ) ) → ( ( 𝑈 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑗 ) → 𝑖 = 𝑗 ) )
25 24 ralrimivva ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) → ∀ 𝑖 ∈ 𝐼 ∀ 𝑗 ∈ 𝐼 ( ( 𝑈 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑗 ) → 𝑖 = 𝑗 ) )
26 dff13 ⊢ ( 𝑈 : 𝐼 –1-1→ 𝐵 ↔ ( 𝑈 : 𝐼 ⟶ 𝐵 ∧ ∀ 𝑖 ∈ 𝐼 ∀ 𝑗 ∈ 𝐼 ( ( 𝑈 ‘ 𝑖 ) = ( 𝑈 ‘ 𝑗 ) → 𝑖 = 𝑗 ) ) )
27 6 25 26 sylanbrc ⊢ ( ( 𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ) → 𝑈 : 𝐼 –1-1→ 𝐵 )