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Theorem resmhm2 17131
Description: One direction of resmhm2b 17132. (Contributed by Mario Carneiro, 18-Jun-2015.)
Hypothesis
Ref Expression
resmhm2.u 𝑈 = (𝑇s 𝑋)
Assertion
Ref Expression
resmhm2 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Proof of Theorem resmhm2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl1 17109 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑈) → 𝑆 ∈ Mnd)
2 submrcl 17117 . . 3 (𝑋 ∈ (SubMnd‘𝑇) → 𝑇 ∈ Mnd)
31, 2anim12i 587 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
4 eqid 2609 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
5 eqid 2609 . . . . 5 (Base‘𝑈) = (Base‘𝑈)
64, 5mhmf 17111 . . . 4 (𝐹 ∈ (𝑆 MndHom 𝑈) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
7 resmhm2.u . . . . . 6 𝑈 = (𝑇s 𝑋)
87submbas 17126 . . . . 5 (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 = (Base‘𝑈))
9 eqid 2609 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
109submss 17121 . . . . 5 (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 ⊆ (Base‘𝑇))
118, 10eqsstr3d 3602 . . . 4 (𝑋 ∈ (SubMnd‘𝑇) → (Base‘𝑈) ⊆ (Base‘𝑇))
12 fss 5954 . . . 4 ((𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ (Base‘𝑈) ⊆ (Base‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
136, 11, 12syl2an 492 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
14 eqid 2609 . . . . . . . 8 (+g𝑆) = (+g𝑆)
15 eqid 2609 . . . . . . . 8 (+g𝑈) = (+g𝑈)
164, 14, 15mhmlin 17113 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
17163expb 1257 . . . . . 6 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
1817adantlr 746 . . . . 5 (((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
19 eqid 2609 . . . . . . . 8 (+g𝑇) = (+g𝑇)
207, 19ressplusg 15766 . . . . . . 7 (𝑋 ∈ (SubMnd‘𝑇) → (+g𝑇) = (+g𝑈))
2120ad2antlr 758 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g𝑇) = (+g𝑈))
2221oveqd 6543 . . . . 5 (((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2318, 22eqtr4d 2646 . . . 4 (((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2423ralrimivva 2953 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
25 eqid 2609 . . . . . 6 (0g𝑆) = (0g𝑆)
26 eqid 2609 . . . . . 6 (0g𝑈) = (0g𝑈)
2725, 26mhm0 17114 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑈) → (𝐹‘(0g𝑆)) = (0g𝑈))
2827adantr 479 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑈))
29 eqid 2609 . . . . . 6 (0g𝑇) = (0g𝑇)
307, 29subm0 17127 . . . . 5 (𝑋 ∈ (SubMnd‘𝑇) → (0g𝑇) = (0g𝑈))
3130adantl 480 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (0g𝑇) = (0g𝑈))
3228, 31eqtr4d 2646 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
3313, 24, 323jca 1234 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇)))
344, 9, 14, 19, 25, 29ismhm 17108 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇))))
353, 33, 34sylanbrc 694 1 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wss 3539  wf 5785  cfv 5789  (class class class)co 6526  Basecbs 15643  s cress 15644  +gcplusg 15716  0gc0g 15871  Mndcmnd 17065   MndHom cmhm 17104  SubMndcsubmnd 17105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-2 10928  df-ndx 15646  df-slot 15647  df-base 15648  df-sets 15649  df-ress 15650  df-plusg 15729  df-0g 15873  df-mgm 17013  df-sgrp 17055  df-mnd 17066  df-mhm 17106  df-submnd 17107
This theorem is referenced by:  resmhm2b  17132  resghm2  17448  zrhpsgnmhm  19696  lgseisenlem4  24847
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