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Theorem mhmco 17185
 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 17162 . . 3 (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝑈 ∈ Mnd)
2 mhmrcl1 17161 . . 3 (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
31, 2anim12ci 589 . 2 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd))
4 eqid 2610 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
5 eqid 2610 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
64, 5mhmf 17163 . . . 4 (𝐹 ∈ (𝑇 MndHom 𝑈) → 𝐹:(Base‘𝑇)⟶(Base‘𝑈))
7 eqid 2610 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
87, 4mhmf 17163 . . . 4 (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
9 fco 5971 . . . 4 ((𝐹:(Base‘𝑇)⟶(Base‘𝑈) ∧ 𝐺:(Base‘𝑆)⟶(Base‘𝑇)) → (𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈))
106, 8, 9syl2an 493 . . 3 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹𝐺):(Base‘𝑆)⟶(Base‘𝑈))
11 eqid 2610 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
12 eqid 2610 . . . . . . . . . 10 (+g𝑇) = (+g𝑇)
137, 11, 12mhmlin 17165 . . . . . . . . 9 ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
14133expb 1258 . . . . . . . 8 ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
1514adantll 746 . . . . . . 7 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
1615fveq2d 6107 . . . . . 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 786 . . . . . . 7 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐹 ∈ (𝑇 MndHom 𝑈))
188ad2antlr 759 . . . . . . . 8 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
19 simprl 790 . . . . . . . 8 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑥 ∈ (Base‘𝑆))
2018, 19ffvelrnd 6268 . . . . . . 7 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺𝑥) ∈ (Base‘𝑇))
21 simprr 792 . . . . . . . 8 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑦 ∈ (Base‘𝑆))
2218, 21ffvelrnd 6268 . . . . . . 7 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐺𝑦) ∈ (Base‘𝑇))
23 eqid 2610 . . . . . . . 8 (+g𝑈) = (+g𝑈)
244, 12, 23mhmlin 17165 . . . . . . 7 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ (𝐺𝑥) ∈ (Base‘𝑇) ∧ (𝐺𝑦) ∈ (Base‘𝑇)) → (𝐹‘((𝐺𝑥)(+g𝑇)(𝐺𝑦))) = ((𝐹‘(𝐺𝑥))(+g𝑈)(𝐹‘(𝐺𝑦))))
2517, 20, 22, 24syl3anc 1318 . . . . . 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 2644 . . . . 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 481 . . . . . . 7 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
287, 11mndcl 17124 . . . . . . . 8 ((𝑆 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
29283expb 1258 . . . . . . 7 ((𝑆 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
3027, 29sylan 487 . . . . . 6 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
31 fvco3 6185 . . . . . 6 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆)) → ((𝐹𝐺)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝐺‘(𝑥(+g𝑆)𝑦))))
3218, 30, 31syl2anc 691 . . . . 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 6185 . . . . . . 7 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3418, 19, 33syl2anc 691 . . . . . 6 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
35 fvco3 6185 . . . . . . 7 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3618, 21, 35syl2anc 691 . . . . . 6 (((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
3734, 36oveq12d 6567 . . . . 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 2654 . . . 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 2954 . . 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 481 . . . . 5 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
41 eqid 2610 . . . . . . 7 (0g𝑆) = (0g𝑆)
427, 41mndidcl 17131 . . . . . 6 (𝑆 ∈ Mnd → (0g𝑆) ∈ (Base‘𝑆))
4327, 42syl 17 . . . . 5 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g𝑆) ∈ (Base‘𝑆))
44 fvco3 6185 . . . . 5 ((𝐺:(Base‘𝑆)⟶(Base‘𝑇) ∧ (0g𝑆) ∈ (Base‘𝑆)) → ((𝐹𝐺)‘(0g𝑆)) = (𝐹‘(𝐺‘(0g𝑆))))
4540, 43, 44syl2anc 691 . . . 4 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹𝐺)‘(0g𝑆)) = (𝐹‘(𝐺‘(0g𝑆))))
46 eqid 2610 . . . . . . 7 (0g𝑇) = (0g𝑇)
4741, 46mhm0 17166 . . . . . 6 (𝐺 ∈ (𝑆 MndHom 𝑇) → (𝐺‘(0g𝑆)) = (0g𝑇))
4847adantl 481 . . . . 5 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐺‘(0g𝑆)) = (0g𝑇))
4948fveq2d 6107 . . . 4 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(𝐺‘(0g𝑆))) = (𝐹‘(0g𝑇)))
50 eqid 2610 . . . . . 6 (0g𝑈) = (0g𝑈)
5146, 50mhm0 17166 . . . . 5 (𝐹 ∈ (𝑇 MndHom 𝑈) → (𝐹‘(0g𝑇)) = (0g𝑈))
5251adantr 480 . . . 4 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑇)) = (0g𝑈))
5345, 49, 523eqtrd 2648 . . 3 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ((𝐹𝐺)‘(0g𝑆)) = (0g𝑈))
5410, 39, 533jca 1235 . 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 17160 . 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 695 1 ((𝐹 ∈ (𝑇 MndHom 𝑈) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MndHom 𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mndcmnd 17117   MndHom cmhm 17156 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158 This theorem is referenced by:  ghmco  17503  rhmco  18560  zrhpsgnmhm  19749  lgseisenlem4  24903
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