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Theorem resmhm2b 17301
Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
Hypothesis
Ref Expression
resmhm2.u 𝑈 = (𝑇s 𝑋)
Assertion
Ref Expression
resmhm2b ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))

Proof of Theorem resmhm2b
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl1 17278 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
21adantl 482 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
3 resmhm2.u . . . . . 6 𝑈 = (𝑇s 𝑋)
43submmnd 17294 . . . . 5 (𝑋 ∈ (SubMnd‘𝑇) → 𝑈 ∈ Mnd)
54ad2antrr 761 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑈 ∈ Mnd)
62, 5jca 554 . . 3 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd))
7 eqid 2621 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
8 eqid 2621 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
97, 8mhmf 17280 . . . . . . . 8 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
109adantl 482 . . . . . . 7 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
11 ffn 6012 . . . . . . 7 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
1210, 11syl 17 . . . . . 6 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
13 simplr 791 . . . . . 6 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ran 𝐹𝑋)
14 df-f 5861 . . . . . 6 (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹𝑋))
1512, 13, 14sylanbrc 697 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
163submbas 17295 . . . . . . 7 (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 = (Base‘𝑈))
1716ad2antrr 761 . . . . . 6 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑋 = (Base‘𝑈))
1817feq3d 5999 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
1915, 18mpbid 222 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
20 eqid 2621 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
21 eqid 2621 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
227, 20, 21mhmlin 17282 . . . . . . . 8 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
23223expb 1263 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2423adantll 749 . . . . . 6 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
253, 21ressplusg 15933 . . . . . . . 8 (𝑋 ∈ (SubMnd‘𝑇) → (+g𝑇) = (+g𝑈))
2625ad3antrrr 765 . . . . . . 7 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g𝑇) = (+g𝑈))
2726oveqd 6632 . . . . . 6 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2824, 27eqtrd 2655 . . . . 5 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2928ralrimivva 2967 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
30 eqid 2621 . . . . . . 7 (0g𝑆) = (0g𝑆)
31 eqid 2621 . . . . . . 7 (0g𝑇) = (0g𝑇)
3230, 31mhm0 17283 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
3332adantl 482 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
343, 31subm0 17296 . . . . . 6 (𝑋 ∈ (SubMnd‘𝑇) → (0g𝑇) = (0g𝑈))
3534ad2antrr 761 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (0g𝑇) = (0g𝑈))
3633, 35eqtrd 2655 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑈))
3719, 29, 363jca 1240 . . 3 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑈)))
38 eqid 2621 . . . 4 (Base‘𝑈) = (Base‘𝑈)
39 eqid 2621 . . . 4 (+g𝑈) = (+g𝑈)
40 eqid 2621 . . . 4 (0g𝑈) = (0g𝑈)
417, 38, 20, 39, 30, 40ismhm 17277 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑈) ↔ ((𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑈))))
426, 37, 41sylanbrc 697 . 2 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑈))
433resmhm2 17300 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
4443ancoms 469 . . 3 ((𝑋 ∈ (SubMnd‘𝑇) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
4544adantlr 750 . 2 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
4642, 45impbida 876 1 ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  wss 3560  ran crn 5085   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  Basecbs 15800  s cress 15801  +gcplusg 15881  0gc0g 16040  Mndcmnd 17234   MndHom cmhm 17273  SubMndcsubmnd 17274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-submnd 17276
This theorem is referenced by:  resghm2b  17618  m2cpmmhm  20490  dchrghm  24915  lgseisenlem4  25037
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