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Theorem subsubmgm 42122
Description: A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)
Hypothesis
Ref Expression
subsubmgm.h 𝐻 = (𝐺s 𝑆)
Assertion
Ref Expression
subsubmgm (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)))

Proof of Theorem subsubmgm
StepHypRef Expression
1 eqid 2651 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
21submgmss 42117 . . . . . . 7 (𝐴 ∈ (SubMgm‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
32adantl 481 . . . . . 6 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4 subsubmgm.h . . . . . . . 8 𝐻 = (𝐺s 𝑆)
54submgmbas 42121 . . . . . . 7 (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 = (Base‘𝐻))
65adantr 480 . . . . . 6 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 = (Base‘𝐻))
73, 6sseqtr4d 3675 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴𝑆)
8 eqid 2651 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
98submgmss 42117 . . . . . 6 (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
109adantr 480 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
117, 10sstrd 3646 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
124oveq1i 6700 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
13 ressabs 15986 . . . . . . 7 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
1412, 13syl5eq 2697 . . . . . 6 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆) → (𝐻s 𝐴) = (𝐺s 𝐴))
157, 14syldan 486 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻s 𝐴) = (𝐺s 𝐴))
16 eqid 2651 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
1716submgmmgm 42120 . . . . . 6 (𝐴 ∈ (SubMgm‘𝐻) → (𝐻s 𝐴) ∈ Mgm)
1817adantl 481 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻s 𝐴) ∈ Mgm)
1915, 18eqeltrrd 2731 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐺s 𝐴) ∈ Mgm)
20 submgmrcl 42107 . . . . . 6 (𝑆 ∈ (SubMgm‘𝐺) → 𝐺 ∈ Mgm)
2120adantr 480 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐺 ∈ Mgm)
22 eqid 2651 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
238, 22issubmgm2 42115 . . . . 5 (𝐺 ∈ Mgm → (𝐴 ∈ (SubMgm‘𝐺) ↔ (𝐴 ⊆ (Base‘𝐺) ∧ (𝐺s 𝐴) ∈ Mgm)))
2421, 23syl 17 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐴 ∈ (SubMgm‘𝐺) ↔ (𝐴 ⊆ (Base‘𝐺) ∧ (𝐺s 𝐴) ∈ Mgm)))
2511, 19, 24mpbir2and 977 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ∈ (SubMgm‘𝐺))
2625, 7jca 553 . 2 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆))
27 simprr 811 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
285adantr 480 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
2927, 28sseqtrd 3674 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3014adantrl 752 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
3122submgmmgm 42120 . . . . 5 (𝐴 ∈ (SubMgm‘𝐺) → (𝐺s 𝐴) ∈ Mgm)
3231ad2antrl 764 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Mgm)
3330, 32eqeltrd 2730 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Mgm)
344submgmmgm 42120 . . . . 5 (𝑆 ∈ (SubMgm‘𝐺) → 𝐻 ∈ Mgm)
3534adantr 480 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Mgm)
361, 16issubmgm2 42115 . . . 4 (𝐻 ∈ Mgm → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ⊆ (Base‘𝐻) ∧ (𝐻s 𝐴) ∈ Mgm)))
3735, 36syl 17 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ⊆ (Base‘𝐻) ∧ (𝐻s 𝐴) ∈ Mgm)))
3829, 33, 37mpbir2and 977 . 2 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubMgm‘𝐻))
3926, 38impbida 895 1 (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wss 3607  cfv 5926  (class class class)co 6690  Basecbs 15904  s cress 15905  Mgmcmgm 17287  SubMgmcsubmgm 42103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mgm 17289  df-submgm 42105
This theorem is referenced by: (None)
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