MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resmhm2b Structured version   Visualization version   GIF version
Theorem resmhm2b 18703
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 18675 . . . 4 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝑆 ∈ Mnd)
21adantl 483 . . 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 18694 . . . 4 (𝑋 ∈ (SubMndβ€˜π‘‡) β†’ π‘ˆ ∈ Mnd)
54ad2antrr 725 . . 3 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ π‘ˆ ∈ Mnd)
6 eqid 2733 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
7 eqid 2733 . . . . . . . . 9 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
86, 7mhmf 18677 . . . . . . . 8 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
98adantl 483 . . . . . . 7 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
109ffnd 6719 . . . . . 6 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹 Fn (Baseβ€˜π‘†))
11 simplr 768 . . . . . 6 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ ran 𝐹 βŠ† 𝑋)
12 df-f 6548 . . . . . 6 (𝐹:(Baseβ€˜π‘†)βŸΆπ‘‹ ↔ (𝐹 Fn (Baseβ€˜π‘†) ∧ ran 𝐹 βŠ† 𝑋))
1310, 11, 12sylanbrc 584 . . . . 5 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹:(Baseβ€˜π‘†)βŸΆπ‘‹)
143submbas 18695 . . . . . . 7 (𝑋 ∈ (SubMndβ€˜π‘‡) β†’ 𝑋 = (Baseβ€˜π‘ˆ))
1514ad2antrr 725 . . . . . 6 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝑋 = (Baseβ€˜π‘ˆ))
1615feq3d 6705 . . . . 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 2733 . . . . . . . . 9 (+gβ€˜π‘†) = (+gβ€˜π‘†)
19 eqid 2733 . . . . . . . . 9 (+gβ€˜π‘‡) = (+gβ€˜π‘‡)
206, 18, 19mhmlin 18679 . . . . . . . 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 1121 . . . . . . 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 713 . . . . . 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 17235 . . . . . . . 8 (𝑋 ∈ (SubMndβ€˜π‘‡) β†’ (+gβ€˜π‘‡) = (+gβ€˜π‘ˆ))
2423ad3antrrr 729 . . . . . . 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 7426 . . . . . 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 2773 . . . . 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 3201 . . . 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 2733 . . . . . . 7 (0gβ€˜π‘†) = (0gβ€˜π‘†)
29 eqid 2733 . . . . . . 7 (0gβ€˜π‘‡) = (0gβ€˜π‘‡)
3028, 29mhm0 18680 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
3130adantl 483 . . . . 5 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
323, 29subm0 18696 . . . . . 6 (𝑋 ∈ (SubMndβ€˜π‘‡) β†’ (0gβ€˜π‘‡) = (0gβ€˜π‘ˆ))
3332ad2antrr 725 . . . . 5 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ (0gβ€˜π‘‡) = (0gβ€˜π‘ˆ))
3431, 33eqtrd 2773 . . . 4 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘ˆ))
3517, 27, 343jca 1129 . . 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 2733 . . . 4 (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘ˆ)
37 eqid 2733 . . . 4 (+gβ€˜π‘ˆ) = (+gβ€˜π‘ˆ)
38 eqid 2733 . . . 4 (0gβ€˜π‘ˆ) = (0gβ€˜π‘ˆ)
396, 36, 18, 37, 28, 38ismhm 18673 . . 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 1345 . 2 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹 ∈ (𝑆 MndHom π‘ˆ))
413resmhm2 18702 . . . 4 ((𝐹 ∈ (𝑆 MndHom π‘ˆ) ∧ 𝑋 ∈ (SubMndβ€˜π‘‡)) β†’ 𝐹 ∈ (𝑆 MndHom 𝑇))
4241ancoms 460 . . 3 ((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ 𝐹 ∈ (𝑆 MndHom π‘ˆ)) β†’ 𝐹 ∈ (𝑆 MndHom 𝑇))
4342adantlr 714 . 2 (((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom π‘ˆ)) β†’ 𝐹 ∈ (𝑆 MndHom 𝑇))
4440, 43impbida 800 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  ran crn 5678   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144   β†Ύs cress 17173  +gcplusg 17197  0gc0g 17385  Mndcmnd 18625   MndHom cmhm 18669  SubMndcsubmnd 18670
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-submnd 18672
This theorem is referenced by:  resghm2b  19110  resrhm2b  20349  m2cpmmhm  22247  dchrghm  26759  lgseisenlem4  26881
  Copyright terms: Public domain W3C validator