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Theorem subsubmgm 18673
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 2725 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
21submgmss 18668 . . . . . . 7 (𝐴 ∈ (SubMgm‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
32adantl 480 . . . . . 6 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4 subsubmgm.h . . . . . . . 8 𝐻 = (𝐺s 𝑆)
54submgmbas 18672 . . . . . . 7 (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 = (Base‘𝐻))
65adantr 479 . . . . . 6 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 = (Base‘𝐻))
73, 6sseqtrrd 4018 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴𝑆)
8 eqid 2725 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
98submgmss 18668 . . . . . 6 (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
109adantr 479 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
117, 10sstrd 3987 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
124oveq1i 7429 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
13 ressabs 17233 . . . . . . 7 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
1412, 13eqtrid 2777 . . . . . 6 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆) → (𝐻s 𝐴) = (𝐺s 𝐴))
157, 14syldan 589 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻s 𝐴) = (𝐺s 𝐴))
16 eqid 2725 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
1716submgmmgm 18671 . . . . . 6 (𝐴 ∈ (SubMgm‘𝐻) → (𝐻s 𝐴) ∈ Mgm)
1817adantl 480 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻s 𝐴) ∈ Mgm)
1915, 18eqeltrrd 2826 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐺s 𝐴) ∈ Mgm)
20 submgmrcl 18658 . . . . . 6 (𝑆 ∈ (SubMgm‘𝐺) → 𝐺 ∈ Mgm)
2120adantr 479 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐺 ∈ Mgm)
22 eqid 2725 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
238, 22issubmgm2 18666 . . . . 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 711 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ∈ (SubMgm‘𝐺))
2625, 7jca 510 . 2 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆))
27 simprr 771 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
285adantr 479 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
2927, 28sseqtrd 4017 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3014adantrl 714 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
3122submgmmgm 18671 . . . . 5 (𝐴 ∈ (SubMgm‘𝐺) → (𝐺s 𝐴) ∈ Mgm)
3231ad2antrl 726 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Mgm)
3330, 32eqeltrd 2825 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Mgm)
344submgmmgm 18671 . . . . 5 (𝑆 ∈ (SubMgm‘𝐺) → 𝐻 ∈ Mgm)
3534adantr 479 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Mgm)
361, 16issubmgm2 18666 . . . 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 711 . 2 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubMgm‘𝐻))
3926, 38impbida 799 1 (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wss 3944  cfv 6549  (class class class)co 7419  Basecbs 17183  s cress 17212  Mgmcmgm 18601  SubMgmcsubmgm 18654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mgm 18603  df-submgm 18656
This theorem is referenced by: (None)
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