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Theorem 0mhm 18590
Description: The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
0mhm.z 0 = (0g𝑁)
0mhm.b 𝐵 = (Base‘𝑀)
Assertion
Ref Expression
0mhm ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))

Proof of Theorem 0mhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd))
2 eqid 2738 . . . . . 6 (Base‘𝑁) = (Base‘𝑁)
3 0mhm.z . . . . . 6 0 = (0g𝑁)
42, 3mndidcl 18531 . . . . 5 (𝑁 ∈ Mnd → 0 ∈ (Base‘𝑁))
54adantl 483 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 ∈ (Base‘𝑁))
6 fconst6g 6729 . . . 4 ( 0 ∈ (Base‘𝑁) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
75, 6syl 17 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
8 simpr 486 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 𝑁 ∈ Mnd)
9 eqid 2738 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
102, 9, 3mndlid 18536 . . . . . . . 8 ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → ( 0 (+g𝑁) 0 ) = 0 )
1110eqcomd 2744 . . . . . . 7 ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → 0 = ( 0 (+g𝑁) 0 ))
128, 4, 11syl2anc2 586 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 = ( 0 (+g𝑁) 0 ))
1312adantr 482 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 0 = ( 0 (+g𝑁) 0 ))
14 0mhm.b . . . . . . . . 9 𝐵 = (Base‘𝑀)
15 eqid 2738 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
1614, 15mndcl 18524 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
17163expb 1121 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
1817adantlr 714 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
193fvexi 6854 . . . . . . 7 0 ∈ V
2019fvconst2 7150 . . . . . 6 ((𝑥(+g𝑀)𝑦) ∈ 𝐵 → ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = 0 )
2118, 20syl 17 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = 0 )
2219fvconst2 7150 . . . . . . 7 (𝑥𝐵 → ((𝐵 × { 0 })‘𝑥) = 0 )
2319fvconst2 7150 . . . . . . 7 (𝑦𝐵 → ((𝐵 × { 0 })‘𝑦) = 0 )
2422, 23oveqan12d 7371 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g𝑁) 0 ))
2524adantl 483 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g𝑁) 0 ))
2613, 21, 253eqtr4d 2788 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)))
2726ralrimivva 3196 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ∀𝑥𝐵𝑦𝐵 ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)))
28 eqid 2738 . . . . . 6 (0g𝑀) = (0g𝑀)
2914, 28mndidcl 18531 . . . . 5 (𝑀 ∈ Mnd → (0g𝑀) ∈ 𝐵)
3029adantr 482 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (0g𝑀) ∈ 𝐵)
3119fvconst2 7150 . . . 4 ((0g𝑀) ∈ 𝐵 → ((𝐵 × { 0 })‘(0g𝑀)) = 0 )
3230, 31syl 17 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ((𝐵 × { 0 })‘(0g𝑀)) = 0 )
337, 27, 323jca 1129 . 2 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ((𝐵 × { 0 }):𝐵⟶(Base‘𝑁) ∧ ∀𝑥𝐵𝑦𝐵 ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) ∧ ((𝐵 × { 0 })‘(0g𝑀)) = 0 ))
3414, 2, 15, 9, 28, 3ismhm 18563 . 2 ((𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁) ↔ ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ ((𝐵 × { 0 }):𝐵⟶(Base‘𝑁) ∧ ∀𝑥𝐵𝑦𝐵 ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) ∧ ((𝐵 × { 0 })‘(0g𝑀)) = 0 )))
351, 33, 34sylanbrc 584 1 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3063  {csn 4585   × cxp 5630  wf 6490  cfv 6494  (class class class)co 7352  Basecbs 17043  +gcplusg 17093  0gc0g 17281  Mndcmnd 18516   MndHom cmhm 18559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-riota 7308  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8726  df-0g 17283  df-mgm 18457  df-sgrp 18506  df-mnd 18517  df-mhm 18561
This theorem is referenced by:  0ghm  18981
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