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Theorem subsubm 17129
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 2610 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
21submss 17122 . . . . . . 7 (𝐴 ∈ (SubMnd‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
32adantl 481 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
4 subsubm.h . . . . . . . 8 𝐻 = (𝐺s 𝑆)
54submbas 17127 . . . . . . 7 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘𝐻))
65adantr 480 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 = (Base‘𝐻))
73, 6sseqtr4d 3605 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴𝑆)
8 eqid 2610 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
98submss 17122 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
109adantr 480 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
117, 10sstrd 3578 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12 eqid 2610 . . . . . . 7 (0g𝐺) = (0g𝐺)
134, 12subm0 17128 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → (0g𝐺) = (0g𝐻))
1413adantr 480 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) = (0g𝐻))
15 eqid 2610 . . . . . . 7 (0g𝐻) = (0g𝐻)
1615subm0cl 17124 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (0g𝐻) ∈ 𝐴)
1716adantl 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐻) ∈ 𝐴)
1814, 17eqeltrd 2688 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) ∈ 𝐴)
194oveq1i 6537 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
20 ressabs 15715 . . . . . . 7 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
2119, 20syl5eq 2656 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆) → (𝐻s 𝐴) = (𝐺s 𝐴))
227, 21syldan 486 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻s 𝐴) = (𝐺s 𝐴))
23 eqid 2610 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
2423submmnd 17126 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (𝐻s 𝐴) ∈ Mnd)
2524adantl 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻s 𝐴) ∈ Mnd)
2622, 25eqeltrrd 2689 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐺s 𝐴) ∈ Mnd)
27 submrcl 17118 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
2827adantr 480 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐺 ∈ Mnd)
29 eqid 2610 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
308, 12, 29issubm2 17120 . . . . 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 1238 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ∈ (SubMnd‘𝐺))
3332, 7jca 553 . 2 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆))
34 simprr 792 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
355adantr 480 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
3634, 35sseqtrd 3604 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3713adantr 480 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) = (0g𝐻))
3812subm0cl 17124 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝐴)
3938ad2antrl 760 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) ∈ 𝐴)
4037, 39eqeltrrd 2689 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐻) ∈ 𝐴)
4121adantrl 748 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
4229submmnd 17126 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (𝐺s 𝐴) ∈ Mnd)
4342ad2antrl 760 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Mnd)
4441, 43eqeltrd 2688 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Mnd)
454submmnd 17126 . . . . 5 (𝑆 ∈ (SubMnd‘𝐺) → 𝐻 ∈ Mnd)
4645adantr 480 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Mnd)
471, 15, 23issubm2 17120 . . . 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 1238 . 2 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubMnd‘𝐻))
5033, 49impbida 873 1 (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wss 3540  cfv 5790  (class class class)co 6527  Basecbs 15644  s cress 15645  0gc0g 15872  Mndcmnd 17066  SubMndcsubmnd 17106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-plusg 15730  df-0g 15874  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-submnd 17108
This theorem is referenced by:  zrhpsgnmhm  19697  amgmlem  24461  nn0archi  28968  amgmwlem  42310  amgmlemALT  42311
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