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Theorem subsubmgm 46177
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 2733 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
21submgmss 46172 . . . . . . 7 (𝐴 ∈ (SubMgm‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
32adantl 483 . . . . . 6 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4 subsubmgm.h . . . . . . . 8 𝐻 = (𝐺s 𝑆)
54submgmbas 46176 . . . . . . 7 (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 = (Base‘𝐻))
65adantr 482 . . . . . 6 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 = (Base‘𝐻))
73, 6sseqtrrd 3986 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴𝑆)
8 eqid 2733 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
98submgmss 46172 . . . . . 6 (𝑆 ∈ (SubMgm‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
109adantr 482 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
117, 10sstrd 3955 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
124oveq1i 7368 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
13 ressabs 17135 . . . . . . 7 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
1412, 13eqtrid 2785 . . . . . 6 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆) → (𝐻s 𝐴) = (𝐺s 𝐴))
157, 14syldan 592 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻s 𝐴) = (𝐺s 𝐴))
16 eqid 2733 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
1716submgmmgm 46175 . . . . . 6 (𝐴 ∈ (SubMgm‘𝐻) → (𝐻s 𝐴) ∈ Mgm)
1817adantl 483 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐻s 𝐴) ∈ Mgm)
1915, 18eqeltrrd 2835 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐺s 𝐴) ∈ Mgm)
20 submgmrcl 46162 . . . . . 6 (𝑆 ∈ (SubMgm‘𝐺) → 𝐺 ∈ Mgm)
2120adantr 482 . . . . 5 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐺 ∈ Mgm)
22 eqid 2733 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
238, 22issubmgm2 46170 . . . . 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 712 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → 𝐴 ∈ (SubMgm‘𝐺))
2625, 7jca 513 . 2 ((𝑆 ∈ (SubMgm‘𝐺) ∧ 𝐴 ∈ (SubMgm‘𝐻)) → (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆))
27 simprr 772 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
285adantr 482 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
2927, 28sseqtrd 3985 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3014adantrl 715 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
3122submgmmgm 46175 . . . . 5 (𝐴 ∈ (SubMgm‘𝐺) → (𝐺s 𝐴) ∈ Mgm)
3231ad2antrl 727 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Mgm)
3330, 32eqeltrd 2834 . . 3 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Mgm)
344submgmmgm 46175 . . . . 5 (𝑆 ∈ (SubMgm‘𝐺) → 𝐻 ∈ Mgm)
3534adantr 482 . . . 4 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Mgm)
361, 16issubmgm2 46170 . . . 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 712 . 2 ((𝑆 ∈ (SubMgm‘𝐺) ∧ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubMgm‘𝐻))
3926, 38impbida 800 1 (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wss 3911  cfv 6497  (class class class)co 7358  Basecbs 17088  s cress 17117  Mgmcmgm 18500  SubMgmcsubmgm 46158
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mgm 18502  df-submgm 46160
This theorem is referenced by: (None)
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