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Theorem subsubm 18829
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 2737 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
21submss 18822 . . . . . . 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 18827 . . . . . . 7 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘𝐻))
65adantr 480 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 = (Base‘𝐻))
73, 6sseqtrrd 4021 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴𝑆)
8 eqid 2737 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
98submss 18822 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
109adantr 480 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
117, 10sstrd 3994 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12 eqid 2737 . . . . . . 7 (0g𝐺) = (0g𝐺)
134, 12subm0 18828 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → (0g𝐺) = (0g𝐻))
1413adantr 480 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) = (0g𝐻))
15 eqid 2737 . . . . . . 7 (0g𝐻) = (0g𝐻)
1615subm0cl 18824 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (0g𝐻) ∈ 𝐴)
1716adantl 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐻) ∈ 𝐴)
1814, 17eqeltrd 2841 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) ∈ 𝐴)
194oveq1i 7441 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
20 ressabs 17294 . . . . . . 7 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
2119, 20eqtrid 2789 . . . . . 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 2737 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
2423submmnd 18826 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (𝐻s 𝐴) ∈ Mnd)
2524adantl 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻s 𝐴) ∈ Mnd)
2622, 25eqeltrrd 2842 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐺s 𝐴) ∈ Mnd)
27 submrcl 18815 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
2827adantr 480 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐺 ∈ Mnd)
29 eqid 2737 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
308, 12, 29issubm2 18817 . . . . 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 1343 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ∈ (SubMnd‘𝐺))
3332, 7jca 511 . 2 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆))
34 simprr 773 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
355adantr 480 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
3634, 35sseqtrd 4020 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3713adantr 480 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) = (0g𝐻))
3812subm0cl 18824 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝐴)
3938ad2antrl 728 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) ∈ 𝐴)
4037, 39eqeltrrd 2842 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐻) ∈ 𝐴)
4121adantrl 716 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
4229submmnd 18826 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (𝐺s 𝐴) ∈ Mnd)
4342ad2antrl 728 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Mnd)
4441, 43eqeltrd 2841 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Mnd)
454submmnd 18826 . . . . 5 (𝑆 ∈ (SubMnd‘𝐺) → 𝐻 ∈ Mnd)
4645adantr 480 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Mnd)
471, 15, 23issubm2 18817 . . . 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 1343 . 2 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubMnd‘𝐻))
5033, 49impbida 801 1 (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  0gc0g 17484  Mndcmnd 18747  SubMndcsubmnd 18795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797
This theorem is referenced by:  zrhpsgnmhm  21602  amgmlem  27033  nn0archi  33375  amgmwlem  49321  amgmlemALT  49322
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