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Theorem 0mhm 17987
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 2824 . . . . . 6 (Base‘𝑁) = (Base‘𝑁)
3 0mhm.z . . . . . 6 0 = (0g𝑁)
42, 3mndidcl 17929 . . . . 5 (𝑁 ∈ Mnd → 0 ∈ (Base‘𝑁))
54adantl 485 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 ∈ (Base‘𝑁))
6 fconst6g 6559 . . . 4 ( 0 ∈ (Base‘𝑁) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
75, 6syl 17 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
8 simpr 488 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 𝑁 ∈ Mnd)
9 eqid 2824 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
102, 9, 3mndlid 17934 . . . . . . . 8 ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → ( 0 (+g𝑁) 0 ) = 0 )
1110eqcomd 2830 . . . . . . 7 ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → 0 = ( 0 (+g𝑁) 0 ))
128, 4, 11syl2anc2 588 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 = ( 0 (+g𝑁) 0 ))
1312adantr 484 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 0 = ( 0 (+g𝑁) 0 ))
14 0mhm.b . . . . . . . . 9 𝐵 = (Base‘𝑀)
15 eqid 2824 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
1614, 15mndcl 17922 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
17163expb 1117 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
1817adantlr 714 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
193fvexi 6676 . . . . . . 7 0 ∈ V
2019fvconst2 6958 . . . . . 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 6958 . . . . . . 7 (𝑥𝐵 → ((𝐵 × { 0 })‘𝑥) = 0 )
2319fvconst2 6958 . . . . . . 7 (𝑦𝐵 → ((𝐵 × { 0 })‘𝑦) = 0 )
2422, 23oveqan12d 7169 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g𝑁) 0 ))
2524adantl 485 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g𝑁) 0 ))
2613, 21, 253eqtr4d 2869 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)))
2726ralrimivva 3186 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ∀𝑥𝐵𝑦𝐵 ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)))
28 eqid 2824 . . . . . 6 (0g𝑀) = (0g𝑀)
2914, 28mndidcl 17929 . . . . 5 (𝑀 ∈ Mnd → (0g𝑀) ∈ 𝐵)
3029adantr 484 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (0g𝑀) ∈ 𝐵)
3119fvconst2 6958 . . . 4 ((0g𝑀) ∈ 𝐵 → ((𝐵 × { 0 })‘(0g𝑀)) = 0 )
3230, 31syl 17 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ((𝐵 × { 0 })‘(0g𝑀)) = 0 )
337, 27, 323jca 1125 . 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 17961 . 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 586 1 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  {csn 4551   × cxp 5541  wf 6340  cfv 6344  (class class class)co 7150  Basecbs 16486  +gcplusg 16568  0gc0g 16716  Mndcmnd 17914   MndHom cmhm 17957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8405  df-0g 16718  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-mhm 17959
This theorem is referenced by:  0ghm  18375
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