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Theorem 0mhm 17580
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 2761 . . . . . 6 (Base‘𝑁) = (Base‘𝑁)
3 0mhm.z . . . . . 6 0 = (0g𝑁)
42, 3mndidcl 17530 . . . . 5 (𝑁 ∈ Mnd → 0 ∈ (Base‘𝑁))
54adantl 473 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 ∈ (Base‘𝑁))
6 fconst6g 6256 . . . 4 ( 0 ∈ (Base‘𝑁) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
75, 6syl 17 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }):𝐵⟶(Base‘𝑁))
8 simpr 479 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 𝑁 ∈ Mnd)
9 eqid 2761 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
102, 9, 3mndlid 17533 . . . . . . . 8 ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → ( 0 (+g𝑁) 0 ) = 0 )
1110eqcomd 2767 . . . . . . 7 ((𝑁 ∈ Mnd ∧ 0 ∈ (Base‘𝑁)) → 0 = ( 0 (+g𝑁) 0 ))
128, 5, 11syl2anc 696 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → 0 = ( 0 (+g𝑁) 0 ))
1312adantr 472 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → 0 = ( 0 (+g𝑁) 0 ))
14 0mhm.b . . . . . . . . 9 𝐵 = (Base‘𝑀)
15 eqid 2761 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
1614, 15mndcl 17523 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
17163expb 1114 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
1817adantlr 753 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
19 fvex 6364 . . . . . . . 8 (0g𝑁) ∈ V
203, 19eqeltri 2836 . . . . . . 7 0 ∈ V
2120fvconst2 6635 . . . . . 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 6635 . . . . . . 7 (𝑥𝐵 → ((𝐵 × { 0 })‘𝑥) = 0 )
2420fvconst2 6635 . . . . . . 7 (𝑦𝐵 → ((𝐵 × { 0 })‘𝑦) = 0 )
2523, 24oveqan12d 6834 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g𝑁) 0 ))
2625adantl 473 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)) = ( 0 (+g𝑁) 0 ))
2713, 22, 263eqtr4d 2805 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)))
2827ralrimivva 3110 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ∀𝑥𝐵𝑦𝐵 ((𝐵 × { 0 })‘(𝑥(+g𝑀)𝑦)) = (((𝐵 × { 0 })‘𝑥)(+g𝑁)((𝐵 × { 0 })‘𝑦)))
29 eqid 2761 . . . . . 6 (0g𝑀) = (0g𝑀)
3014, 29mndidcl 17530 . . . . 5 (𝑀 ∈ Mnd → (0g𝑀) ∈ 𝐵)
3130adantr 472 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (0g𝑀) ∈ 𝐵)
3220fvconst2 6635 . . . 4 ((0g𝑀) ∈ 𝐵 → ((𝐵 × { 0 })‘(0g𝑀)) = 0 )
3331, 32syl 17 . . 3 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → ((𝐵 × { 0 })‘(0g𝑀)) = 0 )
347, 28, 333jca 1123 . 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 17559 . 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 701 1 ((𝑀 ∈ Mnd ∧ 𝑁 ∈ Mnd) → (𝐵 × { 0 }) ∈ (𝑀 MndHom 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2140  wral 3051  Vcvv 3341  {csn 4322   × cxp 5265  wf 6046  cfv 6050  (class class class)co 6815  Basecbs 16080  +gcplusg 16164  0gc0g 16323  Mndcmnd 17516   MndHom cmhm 17555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-map 8028  df-0g 16325  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-mhm 17557
This theorem is referenced by:  0ghm  17896
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