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Theorem subsubm 17966
Description: A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
Hypothesis
Ref Expression
subsubm.h 𝐻 = (𝐺s 𝑆)
Assertion
Ref Expression
subsubm (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)))

Proof of Theorem subsubm
StepHypRef Expression
1 eqid 2826 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
21submss 17959 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
32adantl 482 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4 subsubm.h . . . . . . . 8 𝐻 = (𝐺s 𝑆)
54submbas 17964 . . . . . . 7 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘𝐻))
65adantr 481 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 = (Base‘𝐻))
73, 6sseqtrrd 4012 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴𝑆)
8 eqid 2826 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
98submss 17959 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
109adantr 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
117, 10sstrd 3981 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12 eqid 2826 . . . . . . 7 (0g𝐺) = (0g𝐺)
134, 12subm0 17965 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → (0g𝐺) = (0g𝐻))
1413adantr 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) = (0g𝐻))
15 eqid 2826 . . . . . . 7 (0g𝐻) = (0g𝐻)
1615subm0cl 17961 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (0g𝐻) ∈ 𝐴)
1716adantl 482 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐻) ∈ 𝐴)
1814, 17eqeltrd 2918 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) ∈ 𝐴)
194oveq1i 7158 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
20 ressabs 16553 . . . . . . 7 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
2119, 20syl5eq 2873 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆) → (𝐻s 𝐴) = (𝐺s 𝐴))
227, 21syldan 591 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻s 𝐴) = (𝐺s 𝐴))
23 eqid 2826 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
2423submmnd 17963 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (𝐻s 𝐴) ∈ Mnd)
2524adantl 482 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻s 𝐴) ∈ Mnd)
2622, 25eqeltrrd 2919 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐺s 𝐴) ∈ Mnd)
27 submrcl 17955 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
2827adantr 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐺 ∈ Mnd)
29 eqid 2826 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
308, 12, 29issubm2 17957 . . . . 5 (𝐺 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝐺) ↔ (𝐴 ⊆ (Base‘𝐺) ∧ (0g𝐺) ∈ 𝐴 ∧ (𝐺s 𝐴) ∈ Mnd)))
3128, 30syl 17 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐴 ∈ (SubMnd‘𝐺) ↔ (𝐴 ⊆ (Base‘𝐺) ∧ (0g𝐺) ∈ 𝐴 ∧ (𝐺s 𝐴) ∈ Mnd)))
3211, 18, 26, 31mpbir3and 1336 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ∈ (SubMnd‘𝐺))
3332, 7jca 512 . 2 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆))
34 simprr 769 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
355adantr 481 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
3634, 35sseqtrd 4011 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3713adantr 481 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) = (0g𝐻))
3812subm0cl 17961 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝐴)
3938ad2antrl 724 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) ∈ 𝐴)
4037, 39eqeltrrd 2919 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐻) ∈ 𝐴)
4121adantrl 712 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
4229submmnd 17963 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (𝐺s 𝐴) ∈ Mnd)
4342ad2antrl 724 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Mnd)
4441, 43eqeltrd 2918 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Mnd)
454submmnd 17963 . . . . 5 (𝑆 ∈ (SubMnd‘𝐺) → 𝐻 ∈ Mnd)
4645adantr 481 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Mnd)
471, 15, 23issubm2 17957 . . . 4 (𝐻 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ⊆ (Base‘𝐻) ∧ (0g𝐻) ∈ 𝐴 ∧ (𝐻s 𝐴) ∈ Mnd)))
4846, 47syl 17 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ⊆ (Base‘𝐻) ∧ (0g𝐻) ∈ 𝐴 ∧ (𝐻s 𝐴) ∈ Mnd)))
4936, 40, 44, 48mpbir3and 1336 . 2 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubMnd‘𝐻))
5033, 49impbida 797 1 (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wss 3940  cfv 6352  (class class class)co 7148  Basecbs 16473  s cress 16474  0gc0g 16703  Mndcmnd 17900  SubMndcsubmnd 17943
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-0g 16705  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-submnd 17945
This theorem is referenced by:  zrhpsgnmhm  20644  amgmlem  25481  nn0archi  30830  amgmwlem  44735  amgmlemALT  44736
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