MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subsubm Structured version   Visualization version   GIF version

Theorem subsubm 18673
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 2731 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
21submss 18666 . . . . . . 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 18671 . . . . . . 7 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘𝐻))
65adantr 481 . . . . . 6 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 = (Base‘𝐻))
73, 6sseqtrrd 4019 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴𝑆)
8 eqid 2731 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
98submss 18666 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
109adantr 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
117, 10sstrd 3988 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
12 eqid 2731 . . . . . . 7 (0g𝐺) = (0g𝐺)
134, 12subm0 18672 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → (0g𝐺) = (0g𝐻))
1413adantr 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) = (0g𝐻))
15 eqid 2731 . . . . . . 7 (0g𝐻) = (0g𝐻)
1615subm0cl 18668 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (0g𝐻) ∈ 𝐴)
1716adantl 482 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐻) ∈ 𝐴)
1814, 17eqeltrd 2832 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (0g𝐺) ∈ 𝐴)
194oveq1i 7403 . . . . . . 7 (𝐻s 𝐴) = ((𝐺s 𝑆) ↾s 𝐴)
20 ressabs 17176 . . . . . . 7 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆) → ((𝐺s 𝑆) ↾s 𝐴) = (𝐺s 𝐴))
2119, 20eqtrid 2783 . . . . . 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 2731 . . . . . . 7 (𝐻s 𝐴) = (𝐻s 𝐴)
2423submmnd 18670 . . . . . 6 (𝐴 ∈ (SubMnd‘𝐻) → (𝐻s 𝐴) ∈ Mnd)
2524adantl 482 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐻s 𝐴) ∈ Mnd)
2622, 25eqeltrrd 2833 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → (𝐺s 𝐴) ∈ Mnd)
27 submrcl 18659 . . . . . 6 (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
2827adantr 481 . . . . 5 ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ (SubMnd‘𝐻)) → 𝐺 ∈ Mnd)
29 eqid 2731 . . . . . 6 (𝐺s 𝐴) = (𝐺s 𝐴)
308, 12, 29issubm2 18661 . . . . 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 1342 . . 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 771 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴𝑆)
355adantr 481 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝑆 = (Base‘𝐻))
3634, 35sseqtrd 4018 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ⊆ (Base‘𝐻))
3713adantr 481 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) = (0g𝐻))
3812subm0cl 18668 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝐴)
3938ad2antrl 726 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐺) ∈ 𝐴)
4037, 39eqeltrrd 2833 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (0g𝐻) ∈ 𝐴)
4121adantrl 714 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) = (𝐺s 𝐴))
4229submmnd 18670 . . . . 5 (𝐴 ∈ (SubMnd‘𝐺) → (𝐺s 𝐴) ∈ Mnd)
4342ad2antrl 726 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐺s 𝐴) ∈ Mnd)
4441, 43eqeltrd 2832 . . 3 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → (𝐻s 𝐴) ∈ Mnd)
454submmnd 18670 . . . . 5 (𝑆 ∈ (SubMnd‘𝐺) → 𝐻 ∈ Mnd)
4645adantr 481 . . . 4 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐻 ∈ Mnd)
471, 15, 23issubm2 18661 . . . 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 1342 . 2 ((𝑆 ∈ (SubMnd‘𝐺) ∧ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)) → 𝐴 ∈ (SubMnd‘𝐻))
5033, 49impbida 799 1 (𝑆 ∈ (SubMnd‘𝐺) → (𝐴 ∈ (SubMnd‘𝐻) ↔ (𝐴 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wss 3944  cfv 6532  (class class class)co 7393  Basecbs 17126  s cress 17155  0gc0g 17367  Mndcmnd 18602  SubMndcsubmnd 18646
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-submnd 18648
This theorem is referenced by:  zrhpsgnmhm  21070  amgmlem  26421  nn0archi  32324  amgmwlem  47497  amgmlemALT  47498
  Copyright terms: Public domain W3C validator