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Theorem mhmco 18700
Description: The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
Assertion
Ref Expression
mhmco ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MndHom 𝑈))

Proof of Theorem mhmco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl2 18672 . . 3 (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝑈 ∈ Mnd)
2 mhmrcl1 18671 . . 3 (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
31, 2anim12ci 615 . 2 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd))
4 eqid 2733 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
5 eqid 2733 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
64, 5mhmf 18673 . . . 4 (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
7 eqid 2733 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
87, 4mhmf 18673 . . . 4 (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
9 fco 6738 . . . 4 ((𝐹:(Base‘𝑇)⟶(Base‘𝑈) ∧ 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) → (𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈))
106, 8, 9syl2an 597 . . 3 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈))
11 eqid 2733 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
12 eqid 2733 . . . . . . . . . 10 (+g𝑇) = (+g𝑇)
137, 11, 12mhmlin 18675 . . . . . . . . 9 ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
14133expb 1121 . . . . . . . 8 ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
1514adantll 713 . . . . . . 7 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
1615fveq2d 6892 . . . . . 6 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))) = (𝐹‘((𝐺𝑥)(+g𝑇)(𝐺𝑦))))
17 simpll 766 . . . . . . 7 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐹 ∈ (𝑇 MndHom 𝑈))
188ad2antlr 726 . . . . . . . 8 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
19 simprl 770 . . . . . . . 8 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
2018, 19ffvelcdmd 7083 . . . . . . 7 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺𝑥) ∈ (Base‘𝑇))
21 simprr 772 . . . . . . . 8 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
2218, 21ffvelcdmd 7083 . . . . . . 7 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺𝑦) ∈ (Base‘𝑇))
23 eqid 2733 . . . . . . . 8 (+g𝑈) = (+g𝑈)
244, 12, 23mhmlin 18675 . . . . . . 7 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ (𝐺𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑦) ∈ (Base‘𝑇)) → (𝐹‘((𝐺𝑥)(+g𝑇)(𝐺𝑦))) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
2517, 20, 22, 24syl3anc 1372 . . . . . 6 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘((𝐺𝑥)(+g𝑇)(𝐺𝑦))) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
2616, 25eqtrd 2773 . . . . 5 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
272adantl 483 . . . . . . 7 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
287, 11mndcl 18629 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
29283expb 1121 . . . . . . 7 ((𝑆 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
3027, 29sylan 581 . . . . . 6 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
31 fvco3 6986 . . . . . 6 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆)) → ((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))))
3218, 30, 31syl2anc 585 . . . . 5 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))))
33 fvco3 6986 . . . . . . 7 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3418, 19, 33syl2anc 585 . . . . . 6 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
35 fvco3 6986 . . . . . . 7 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3618, 21, 35syl2anc 585 . . . . . 6 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3734, 36oveq12d 7422 . . . . 5 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦)) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
3826, 32, 373eqtr4d 2783 . . . 4 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦)))
3938ralrimivva 3201 . . 3 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦)))
408adantl 483 . . . . 5 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
41 eqid 2733 . . . . . . 7 (0g𝑆) = (0g𝑆)
427, 41mndidcl 18636 . . . . . 6 (𝑆 ∈ Mnd → (0g𝑆) ∈ (Base‘𝑆))
4327, 42syl 17 . . . . 5 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g𝑆) ∈ (Base‘𝑆))
44 fvco3 6986 . . . . 5 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (0g𝑆) ∈ (Base‘𝑆)) → ((𝐹𝐺)‘(0g𝑆)) = (𝐹‘(𝐺‘(0g𝑆))))
4540, 43, 44syl2anc 585 . . . 4 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹𝐺)‘(0g𝑆)) = (𝐹‘(𝐺‘(0g𝑆))))
46 eqid 2733 . . . . . . 7 (0g𝑇) = (0g𝑇)
4741, 46mhm0 18676 . . . . . 6 (𝐺 ∈ (𝑆 MndHom 𝑇) → (𝐺‘(0g𝑆)) = (0g𝑇))
4847adantl 483 . . . . 5 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐺‘(0g𝑆)) = (0g𝑇))
4948fveq2d 6892 . . . 4 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(𝐺‘(0g𝑆))) = (𝐹‘(0g𝑇)))
50 eqid 2733 . . . . . 6 (0g𝑈) = (0g𝑈)
5146, 50mhm0 18676 . . . . 5 (𝐹 ∈ (𝑇 MndHom 𝑈) → (𝐹‘(0g𝑇)) = (0g𝑈))
5251adantr 482 . . . 4 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑇)) = (0g𝑈))
5345, 49, 523eqtrd 2777 . . 3 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹𝐺)‘(0g𝑆)) = (0g𝑈))
5410, 39, 533jca 1129 . 2 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦)) ∧ ((𝐹𝐺)‘(0g𝑆)) = (0g𝑈)))
557, 5, 11, 23, 41, 50ismhm 18669 . 2 ((𝐹𝐺) ∈ (𝑆 MndHom 𝑈) ↔ ((𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd) ∧ ((𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝐺)‘𝑥)(+g𝑈)((𝐹𝐺)‘𝑦)) ∧ ((𝐹𝐺)‘(0g𝑆)) = (0g𝑈))))
563, 54, 55sylanbrc 584 1 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MndHom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  ccom 5679  wf 6536  cfv 6540  (class class class)co 7404  Basecbs 17140  +gcplusg 17193  0gc0g 17381  Mndcmnd 18621   MndHom cmhm 18665
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 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8818  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667
This theorem is referenced by:  ghmco  19106  rhmco  20265  zrhpsgnmhm  21121  lgseisenlem4  26861
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