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Theorem 0mhm 17298
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 2621 . . . . . 6 (Base‘𝑁) = (Base‘𝑁)
3 0mhm.z . . . . . 6 0 = (0g𝑁)
42, 3mndidcl 17248 . . . . 5 (𝑁 ∈ Mnd → 0 ∈ (Base‘𝑁))
54adantl 482 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 ∈ (Base‘𝑁))
6 fconst6g 6061 . . . 4 ( 0 ∈ (Base‘𝑁) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
75, 6syl 17 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
8 simpr 477 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 𝑁 ∈ Mnd)
9 eqid 2621 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
102, 9, 3mndlid 17251 . . . . . . . 8 ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → ( 0 (+g𝑁) 0 ) = 0 )
1110eqcomd 2627 . . . . . . 7 ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → 0 = ( 0 (+g𝑁) 0 ))
128, 5, 11syl2anc 692 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 = ( 0 (+g𝑁) 0 ))
1312adantr 481 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 0 = ( 0 (+g𝑁) 0 ))
14 0mhm.b . . . . . . . . 9 𝐵 = (Base‘𝑀)
15 eqid 2621 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
1614, 15mndcl 17241 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
17163expb 1263 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
1817adantlr 750 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
19 fvex 6168 . . . . . . . 8 (0g𝑁) ∈ V
203, 19eqeltri 2694 . . . . . . 7 0 ∈ V
2120fvconst2 6434 . . . . . 6 ((𝑥(+g𝑀)𝑦) ∈ 𝐵 → ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = 0 )
2218, 21syl 17 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = 0 )
2320fvconst2 6434 . . . . . . 7 (𝑥𝐵 → ((𝐵 × { 0 })‘𝑥) = 0 )
2420fvconst2 6434 . . . . . . 7 (𝑦𝐵 → ((𝐵 × { 0 })‘𝑦) = 0 )
2523, 24oveqan12d 6634 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g𝑁) 0 ))
2625adantl 482 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g𝑁) 0 ))
2713, 22, 263eqtr4d 2665 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)))
2827ralrimivva 2967 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ∀𝑥𝐵𝑦𝐵 ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)))
29 eqid 2621 . . . . . 6 (0g𝑀) = (0g𝑀)
3014, 29mndidcl 17248 . . . . 5 (𝑀 ∈ Mnd → (0g𝑀) ∈ 𝐵)
3130adantr 481 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (0g𝑀) ∈ 𝐵)
3220fvconst2 6434 . . . 4 ((0g𝑀) ∈ 𝐵 → ((𝐵 × { 0 })‘(0g𝑀)) = 0 )
3331, 32syl 17 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ((𝐵 × { 0 })‘(0g𝑀)) = 0 )
347, 28, 333jca 1240 . 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 ))
3514, 2, 15, 9, 29, 3ismhm 17277 . 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 )))
361, 34, 35sylanbrc 697 1 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  {csn 4155   × cxp 5082  wf 5853  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  0gc0g 16040  Mndcmnd 17234   MndHom cmhm 17273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275
This theorem is referenced by:  0ghm  17614
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