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Theorem subsubm 18052
 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 2758 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
21submss 18045 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
32adantl 485 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4 subsubm.h . . . . . . . 8 𝐻 = (𝐺s 𝑆)
54submbas 18050 . . . . . . 7 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘𝐻))
65adantr 484 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 = (Base‘𝐻))
73, 6sseqtrrd 3935 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴𝑆)
8 eqid 2758 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
98submss 18045 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
109adantr 484 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
117, 10sstrd 3904 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12 eqid 2758 . . . . . . 7 (0g𝐺) = (0g𝐺)
134, 12subm0 18051 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → (0g𝐺) = (0g𝐻))
1413adantr 484 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) = (0g𝐻))
15 eqid 2758 . . . . . . 7 (0g𝐻) = (0g𝐻)
1615subm0cl 18047 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (0g𝐻) ∈ 𝐴)
1716adantl 485 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐻) ∈ 𝐴)
1814, 17eqeltrd 2852 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) ∈ 𝐴)
194oveq1i 7165 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
20 ressabs 16626 . . . . . . 7 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
2119, 20syl5eq 2805 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆) → (𝐻s 𝐴) = (𝐺s 𝐴))
227, 21syldan 594 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻s 𝐴) = (𝐺s 𝐴))
23 eqid 2758 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
2423submmnd 18049 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (𝐻s 𝐴) ∈ Mnd)
2524adantl 485 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻s 𝐴) ∈ Mnd)
2622, 25eqeltrrd 2853 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐺s 𝐴) ∈ Mnd)
27 submrcl 18038 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
2827adantr 484 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐺 ∈ Mnd)
29 eqid 2758 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
308, 12, 29issubm2 18040 . . . . 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 1339 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ∈ (SubMnd‘𝐺))
3332, 7jca 515 . 2 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆))
34 simprr 772 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
355adantr 484 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
3634, 35sseqtrd 3934 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3713adantr 484 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) = (0g𝐻))
3812subm0cl 18047 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝐴)
3938ad2antrl 727 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) ∈ 𝐴)
4037, 39eqeltrrd 2853 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐻) ∈ 𝐴)
4121adantrl 715 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
4229submmnd 18049 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (𝐺s 𝐴) ∈ Mnd)
4342ad2antrl 727 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Mnd)
4441, 43eqeltrd 2852 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Mnd)
454submmnd 18049 . . . . 5 (𝑆 ∈ (SubMnd‘𝐺) → 𝐻 ∈ Mnd)
4645adantr 484 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Mnd)
471, 15, 23issubm2 18040 . . . 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 1339 . 2 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubMnd‘𝐻))
5033, 49impbida 800 1 (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860  ‘cfv 6339  (class class class)co 7155  Basecbs 16546   ↾s cress 16547  0gc0g 16776  Mndcmnd 17982  SubMndcsubmnd 18026 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-er 8304  df-en 8533  df-dom 8534  df-sdom 8535  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-ndx 16549  df-slot 16550  df-base 16552  df-sets 16553  df-ress 16554  df-plusg 16641  df-0g 16778  df-mgm 17923  df-sgrp 17972  df-mnd 17983  df-submnd 18028 This theorem is referenced by:  zrhpsgnmhm  20354  amgmlem  25679  nn0archi  31072  amgmwlem  45794  amgmlemALT  45795
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