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Metamath Proof Explorer

Theorem imasmnd

Description: The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015)

Ref Expression
Hypotheses imasmnd.u ⊢ ( 𝜑 → 𝑈 = ( 𝐹 “s 𝑅 ) )
imasmnd.v ⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) )
imasmnd.p ⊢ + = ( +g ‘ 𝑅 )
imasmnd.f ⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 )
imasmnd.e ⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑝 ) ∧ ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑞 ) ) → ( 𝐹 ‘ ( 𝑎 + 𝑏 ) ) = ( 𝐹 ‘ ( 𝑝 + 𝑞 ) ) ) )
imasmnd.r ⊢ ( 𝜑 → 𝑅 ∈ Mnd )
imasmnd.z ⊢ 0 = ( 0g ‘ 𝑅 )
Assertion imasmnd ( 𝜑 → ( 𝑈 ∈ Mnd ∧ ( 𝐹 ‘ 0 ) = ( 0g ‘ 𝑈 ) ) )

Proof

Step Hyp Ref Expression
1 imasmnd.u ⊢ ( 𝜑 → 𝑈 = ( 𝐹 “s 𝑅 ) )
2 imasmnd.v ⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) )
3 imasmnd.p ⊢ + = ( +g ‘ 𝑅 )
4 imasmnd.f ⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 )
5 imasmnd.e ⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑝 ) ∧ ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑞 ) ) → ( 𝐹 ‘ ( 𝑎 + 𝑏 ) ) = ( 𝐹 ‘ ( 𝑝 + 𝑞 ) ) ) )
6 imasmnd.r ⊢ ( 𝜑 → 𝑅 ∈ Mnd )
7 imasmnd.z ⊢ 0 = ( 0g ‘ 𝑅 )
8 6 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑅 ∈ Mnd )
9 simp2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑥 ∈ 𝑉 )
10 2 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑉 = ( Base ‘ 𝑅 ) )
11 9 10 eleqtrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑥 ∈ ( Base ‘ 𝑅 ) )
12 simp3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑦 ∈ 𝑉 )
13 12 10 eleqtrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → 𝑦 ∈ ( Base ‘ 𝑅 ) )
14 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
15 14 3 mndcl ⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 + 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
16 8 11 13 15 syl3anc ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑥 + 𝑦 ) ∈ ( Base ‘ 𝑅 ) )
17 16 10 eleqtrrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑥 + 𝑦 ) ∈ 𝑉 )
18 6 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑅 ∈ Mnd )
19 11 3adant3r3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) )
20 13 3adant3r3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑦 ∈ ( Base ‘ 𝑅 ) )
21 simpr3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑧 ∈ 𝑉 )
22 2 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑉 = ( Base ‘ 𝑅 ) )
23 21 22 eleqtrd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → 𝑧 ∈ ( Base ‘ 𝑅 ) )
24 14 3 mndass ⊢ ( ( 𝑅 ∈ Mnd ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑧 ∈ ( Base ‘ 𝑅 ) ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
25 18 19 20 23 24 syl13anc ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
26 25 fveq2d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝐹 ‘ ( ( 𝑥 + 𝑦 ) + 𝑧 ) ) = ( 𝐹 ‘ ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) )
27 14 7 mndidcl ⊢ ( 𝑅 ∈ Mnd → 0 ∈ ( Base ‘ 𝑅 ) )
28 6 27 syl ⊢ ( 𝜑 → 0 ∈ ( Base ‘ 𝑅 ) )
29 28 2 eleqtrrd ⊢ ( 𝜑 → 0 ∈ 𝑉 )
30 2 eleq2d ⊢ ( 𝜑 → ( 𝑥 ∈ 𝑉 ↔ 𝑥 ∈ ( Base ‘ 𝑅 ) ) )
31 30 biimpa ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → 𝑥 ∈ ( Base ‘ 𝑅 ) )
32 14 3 7 mndlid ⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 0 + 𝑥 ) = 𝑥 )
33 6 31 32 syl2an2r ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 0 + 𝑥 ) = 𝑥 )
34 33 fveq2d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝐹 ‘ ( 0 + 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) )
35 14 3 7 mndrid ⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝑥 + 0 ) = 𝑥 )
36 6 31 35 syl2an2r ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝑥 + 0 ) = 𝑥 )
37 36 fveq2d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑉 ) → ( 𝐹 ‘ ( 𝑥 + 0 ) ) = ( 𝐹 ‘ 𝑥 ) )
38 1 2 3 4 5 6 17 26 29 34 37 imasmnd2 ⊢ ( 𝜑 → ( 𝑈 ∈ Mnd ∧ ( 𝐹 ‘ 0 ) = ( 0g ‘ 𝑈 ) ) )