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Theorem resmgmhm 46082
Description: Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
Hypothesis
Ref Expression
resmgmhm.u 𝑈 = (𝑆s 𝑋)
Assertion
Ref Expression
resmgmhm ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MgmHom 𝑇))

Proof of Theorem resmgmhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmrcl 46065 . . . 4 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
21simprd 496 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑇 ∈ Mgm)
3 resmgmhm.u . . . 4 𝑈 = (𝑆s 𝑋)
43submgmmgm 46079 . . 3 (𝑋 ∈ (SubMgm‘𝑆) → 𝑈 ∈ Mgm)
52, 4anim12ci 614 . 2 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝑈 ∈ Mgm ∧ 𝑇 ∈ Mgm))
6 eqid 2736 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
7 eqid 2736 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
86, 7mgmhmf 46068 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
96submgmss 46076 . . . . 5 (𝑋 ∈ (SubMgm‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
10 fssres 6708 . . . . 5 ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
118, 9, 10syl2an 596 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
129adantl 482 . . . . . 6 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
133, 6ressbas2 17120 . . . . . 6 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
1412, 13syl 17 . . . . 5 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → 𝑋 = (Base‘𝑈))
1514feq2d 6654 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ((𝐹𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
1611, 15mpbid 231 . . 3 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇))
17 simpll 765 . . . . . . 7 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
189ad2antlr 725 . . . . . . . 8 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑋 ⊆ (Base‘𝑆))
19 simprl 769 . . . . . . . 8 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
2018, 19sseldd 3945 . . . . . . 7 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥 ∈ (Base‘𝑆))
21 simprr 771 . . . . . . . 8 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
2218, 21sseldd 3945 . . . . . . 7 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦 ∈ (Base‘𝑆))
23 eqid 2736 . . . . . . . 8 (+g𝑆) = (+g𝑆)
24 eqid 2736 . . . . . . . 8 (+g𝑇) = (+g𝑇)
256, 23, 24mgmhmlin 46070 . . . . . . 7 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2617, 20, 22, 25syl3anc 1371 . . . . . 6 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2723submgmcl 46078 . . . . . . . . 9 ((𝑋 ∈ (SubMgm‘𝑆) ∧ 𝑥𝑋𝑦𝑋) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
28273expb 1120 . . . . . . . 8 ((𝑋 ∈ (SubMgm‘𝑆) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
2928adantll 712 . . . . . . 7 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
30 fvres 6861 . . . . . . 7 ((𝑥(+g𝑆)𝑦) ∈ 𝑋 → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
3129, 30syl 17 . . . . . 6 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
32 fvres 6861 . . . . . . . 8 (𝑥𝑋 → ((𝐹𝑋)‘𝑥) = (𝐹𝑥))
33 fvres 6861 . . . . . . . 8 (𝑦𝑋 → ((𝐹𝑋)‘𝑦) = (𝐹𝑦))
3432, 33oveqan12d 7376 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3534adantl 482 . . . . . 6 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3626, 31, 353eqtr4d 2786 . . . . 5 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
3736ralrimivva 3197 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
383, 23ressplusg 17171 . . . . . . . . 9 (𝑋 ∈ (SubMgm‘𝑆) → (+g𝑆) = (+g𝑈))
3938adantl 482 . . . . . . . 8 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (+g𝑆) = (+g𝑈))
4039oveqd 7374 . . . . . . 7 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝑥(+g𝑆)𝑦) = (𝑥(+g𝑈)𝑦))
4140fveqeq2d 6850 . . . . . 6 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ↔ ((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
4214, 41raleqbidv 3319 . . . . 5 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (∀𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
4314, 42raleqbidv 3319 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (∀𝑥𝑋𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
4437, 43mpbid 231 . . 3 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
4516, 44jca 512 . 2 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ((𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
46 eqid 2736 . . 3 (Base‘𝑈) = (Base‘𝑈)
47 eqid 2736 . . 3 (+g𝑈) = (+g𝑈)
4846, 7, 47, 24ismgmhm 46067 . 2 ((𝐹𝑋) ∈ (𝑈 MgmHom 𝑇) ↔ ((𝑈 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ ((𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))))
495, 45, 48sylanbrc 583 1 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MgmHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  wss 3910  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  Basecbs 17083  s cress 17112  +gcplusg 17133  Mgmcmgm 18495   MgmHom cmgmhm 46061  SubMgmcsubmgm 46062
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mgm 18497  df-mgmhm 46063  df-submgm 46064
This theorem is referenced by: (None)
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