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Theorem resmhm2b 18739
Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
Hypothesis
Ref Expression
resmhm2.u π‘ˆ = (𝑇 β†Ύs 𝑋)
Assertion
Ref Expression
resmhm2b ((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
Proof of Theorem resmhm2b
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl1 18709 . . . 4 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝑆 ∈ Mnd)
21adantl 480 . . 3 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝑆 ∈ Mnd)
3 resmhm2.u . . . . 5 π‘ˆ = (𝑇 β†Ύs 𝑋)
43submmnd 18730 . . . 4 (𝑋 ∈ (SubMndβ€˜π‘‡) β†’ π‘ˆ ∈ Mnd)
54ad2antrr 722 . . 3 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ π‘ˆ ∈ Mnd)
6 eqid 2730 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
7 eqid 2730 . . . . . . . . 9 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
86, 7mhmf 18711 . . . . . . . 8 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
98adantl 480 . . . . . . 7 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
109ffnd 6717 . . . . . 6 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹 Fn (Baseβ€˜π‘†))
11 simplr 765 . . . . . 6 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ ran 𝐹 βŠ† 𝑋)
12 df-f 6546 . . . . . 6 (𝐹:(Baseβ€˜π‘†)βŸΆπ‘‹ ↔ (𝐹 Fn (Baseβ€˜π‘†) ∧ ran 𝐹 βŠ† 𝑋))
1310, 11, 12sylanbrc 581 . . . . 5 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹:(Baseβ€˜π‘†)βŸΆπ‘‹)
143submbas 18731 . . . . . . 7 (𝑋 ∈ (SubMndβ€˜π‘‡) β†’ 𝑋 = (Baseβ€˜π‘ˆ))
1514ad2antrr 722 . . . . . 6 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝑋 = (Baseβ€˜π‘ˆ))
1615feq3d 6703 . . . . 5 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ (𝐹:(Baseβ€˜π‘†)βŸΆπ‘‹ ↔ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘ˆ)))
1713, 16mpbid 231 . . . 4 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘ˆ))
18 eqid 2730 . . . . . . . . 9 (+gβ€˜π‘†) = (+gβ€˜π‘†)
19 eqid 2730 . . . . . . . . 9 (+gβ€˜π‘‡) = (+gβ€˜π‘‡)
206, 18, 19mhmlin 18715 . . . . . . . 8 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
21203expb 1118 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†))) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
2221adantll 710 . . . . . 6 ((((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†))) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
233, 19ressplusg 17239 . . . . . . . 8 (𝑋 ∈ (SubMndβ€˜π‘‡) β†’ (+gβ€˜π‘‡) = (+gβ€˜π‘ˆ))
2423ad3antrrr 726 . . . . . . 7 ((((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†))) β†’ (+gβ€˜π‘‡) = (+gβ€˜π‘ˆ))
2524oveqd 7428 . . . . . 6 ((((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†))) β†’ ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘ˆ)(πΉβ€˜π‘¦)))
2622, 25eqtrd 2770 . . . . 5 ((((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†))) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘ˆ)(πΉβ€˜π‘¦)))
2726ralrimivva 3198 . . . 4 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ βˆ€π‘₯ ∈ (Baseβ€˜π‘†)βˆ€π‘¦ ∈ (Baseβ€˜π‘†)(πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘ˆ)(πΉβ€˜π‘¦)))
28 eqid 2730 . . . . . . 7 (0gβ€˜π‘†) = (0gβ€˜π‘†)
29 eqid 2730 . . . . . . 7 (0gβ€˜π‘‡) = (0gβ€˜π‘‡)
3028, 29mhm0 18716 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
3130adantl 480 . . . . 5 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
323, 29subm0 18732 . . . . . 6 (𝑋 ∈ (SubMndβ€˜π‘‡) β†’ (0gβ€˜π‘‡) = (0gβ€˜π‘ˆ))
3332ad2antrr 722 . . . . 5 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ (0gβ€˜π‘‡) = (0gβ€˜π‘ˆ))
3431, 33eqtrd 2770 . . . 4 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘ˆ))
3517, 27, 343jca 1126 . . 3 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ (𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘ˆ) ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘†)βˆ€π‘¦ ∈ (Baseβ€˜π‘†)(πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘ˆ)(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘ˆ)))
36 eqid 2730 . . . 4 (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘ˆ)
37 eqid 2730 . . . 4 (+gβ€˜π‘ˆ) = (+gβ€˜π‘ˆ)
38 eqid 2730 . . . 4 (0gβ€˜π‘ˆ) = (0gβ€˜π‘ˆ)
396, 36, 18, 37, 28, 38ismhm 18707 . . 3 (𝐹 ∈ (𝑆 MndHom π‘ˆ) ↔ ((𝑆 ∈ Mnd ∧ π‘ˆ ∈ Mnd) ∧ (𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘ˆ) ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘†)βˆ€π‘¦ ∈ (Baseβ€˜π‘†)(πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘ˆ)(πΉβ€˜π‘¦)) ∧ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘ˆ))))
402, 5, 35, 39syl21anbrc 1342 . 2 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹 ∈ (𝑆 MndHom π‘ˆ))
413resmhm2 18738 . . . 4 ((𝐹 ∈ (𝑆 MndHom π‘ˆ) ∧ 𝑋 ∈ (SubMndβ€˜π‘‡)) β†’ 𝐹 ∈ (𝑆 MndHom 𝑇))
4241ancoms 457 . . 3 ((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ 𝐹 ∈ (𝑆 MndHom π‘ˆ)) β†’ 𝐹 ∈ (𝑆 MndHom 𝑇))
4342adantlr 711 . 2 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom π‘ˆ)) β†’ 𝐹 ∈ (𝑆 MndHom 𝑇))
4440, 43impbida 797 1 ((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βŠ† wss 3947  ran crn 5676   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148   β†Ύs cress 17177  +gcplusg 17201  0gc0g 17389  Mndcmnd 18659   MndHom cmhm 18703  SubMndcsubmnd 18704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  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-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706
This theorem is referenced by:  resghm2b  19148  resrhm2b  20492  m2cpmmhm  22467  dchrghm  26995  lgseisenlem4  27117
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