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Theorem subsubm 18697
Description: A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
Hypothesis
Ref Expression
subsubm.h 𝐻 = (𝐺 β†Ύs 𝑆)
Assertion
Ref Expression
subsubm (𝑆 ∈ (SubMndβ€˜πΊ) β†’ (𝐴 ∈ (SubMndβ€˜π») ↔ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)))
Proof of Theorem subsubm
StepHypRef Expression
1 eqid 2733 . . . . . . . 8 (Baseβ€˜π») = (Baseβ€˜π»)
21submss 18690 . . . . . . 7 (𝐴 ∈ (SubMndβ€˜π») β†’ 𝐴 βŠ† (Baseβ€˜π»))
32adantl 483 . . . . . 6 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† (Baseβ€˜π»))
4 subsubm.h . . . . . . . 8 𝐻 = (𝐺 β†Ύs 𝑆)
54submbas 18695 . . . . . . 7 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝑆 = (Baseβ€˜π»))
65adantr 482 . . . . . 6 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝑆 = (Baseβ€˜π»))
73, 6sseqtrrd 4024 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† 𝑆)
8 eqid 2733 . . . . . . 7 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
98submss 18690 . . . . . 6 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
109adantr 482 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
117, 10sstrd 3993 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† (Baseβ€˜πΊ))
12 eqid 2733 . . . . . . 7 (0gβ€˜πΊ) = (0gβ€˜πΊ)
134, 12subm0 18696 . . . . . 6 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ (0gβ€˜πΊ) = (0gβ€˜π»))
1413adantr 482 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (0gβ€˜πΊ) = (0gβ€˜π»))
15 eqid 2733 . . . . . . 7 (0gβ€˜π») = (0gβ€˜π»)
1615subm0cl 18692 . . . . . 6 (𝐴 ∈ (SubMndβ€˜π») β†’ (0gβ€˜π») ∈ 𝐴)
1716adantl 483 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (0gβ€˜π») ∈ 𝐴)
1814, 17eqeltrd 2834 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (0gβ€˜πΊ) ∈ 𝐴)
194oveq1i 7419 . . . . . . 7 (𝐻 β†Ύs 𝐴) = ((𝐺 β†Ύs 𝑆) β†Ύs 𝐴)
20 ressabs 17194 . . . . . . 7 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆) β†’ ((𝐺 β†Ύs 𝑆) β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
2119, 20eqtrid 2785 . . . . . 6 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆) β†’ (𝐻 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
227, 21syldan 592 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐻 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
23 eqid 2733 . . . . . . 7 (𝐻 β†Ύs 𝐴) = (𝐻 β†Ύs 𝐴)
2423submmnd 18694 . . . . . 6 (𝐴 ∈ (SubMndβ€˜π») β†’ (𝐻 β†Ύs 𝐴) ∈ Mnd)
2524adantl 483 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐻 β†Ύs 𝐴) ∈ Mnd)
2622, 25eqeltrrd 2835 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐺 β†Ύs 𝐴) ∈ Mnd)
27 submrcl 18683 . . . . . 6 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝐺 ∈ Mnd)
2827adantr 482 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐺 ∈ Mnd)
29 eqid 2733 . . . . . 6 (𝐺 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴)
308, 12, 29issubm2 18685 . . . . 5 (𝐺 ∈ Mnd β†’ (𝐴 ∈ (SubMndβ€˜πΊ) ↔ (𝐴 βŠ† (Baseβ€˜πΊ) ∧ (0gβ€˜πΊ) ∈ 𝐴 ∧ (𝐺 β†Ύs 𝐴) ∈ Mnd)))
3128, 30syl 17 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐴 ∈ (SubMndβ€˜πΊ) ↔ (𝐴 βŠ† (Baseβ€˜πΊ) ∧ (0gβ€˜πΊ) ∈ 𝐴 ∧ (𝐺 β†Ύs 𝐴) ∈ Mnd)))
3211, 18, 26, 31mpbir3and 1343 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 ∈ (SubMndβ€˜πΊ))
3332, 7jca 513 . 2 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆))
34 simprr 772 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐴 βŠ† 𝑆)
355adantr 482 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝑆 = (Baseβ€˜π»))
3634, 35sseqtrd 4023 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐴 βŠ† (Baseβ€˜π»))
3713adantr 482 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (0gβ€˜πΊ) = (0gβ€˜π»))
3812subm0cl 18692 . . . . 5 (𝐴 ∈ (SubMndβ€˜πΊ) β†’ (0gβ€˜πΊ) ∈ 𝐴)
3938ad2antrl 727 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (0gβ€˜πΊ) ∈ 𝐴)
4037, 39eqeltrrd 2835 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (0gβ€˜π») ∈ 𝐴)
4121adantrl 715 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐻 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
4229submmnd 18694 . . . . 5 (𝐴 ∈ (SubMndβ€˜πΊ) β†’ (𝐺 β†Ύs 𝐴) ∈ Mnd)
4342ad2antrl 727 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐺 β†Ύs 𝐴) ∈ Mnd)
4441, 43eqeltrd 2834 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐻 β†Ύs 𝐴) ∈ Mnd)
454submmnd 18694 . . . . 5 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝐻 ∈ Mnd)
4645adantr 482 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐻 ∈ Mnd)
471, 15, 23issubm2 18685 . . . 4 (𝐻 ∈ Mnd β†’ (𝐴 ∈ (SubMndβ€˜π») ↔ (𝐴 βŠ† (Baseβ€˜π») ∧ (0gβ€˜π») ∈ 𝐴 ∧ (𝐻 β†Ύs 𝐴) ∈ Mnd)))
4846, 47syl 17 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐴 ∈ (SubMndβ€˜π») ↔ (𝐴 βŠ† (Baseβ€˜π») ∧ (0gβ€˜π») ∈ 𝐴 ∧ (𝐻 β†Ύs 𝐴) ∈ Mnd)))
4936, 40, 44, 48mpbir3and 1343 . 2 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐴 ∈ (SubMndβ€˜π»))
5033, 49impbida 800 1 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ (𝐴 ∈ (SubMndβ€˜π») ↔ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144   β†Ύs cress 17173  0gc0g 17385  Mndcmnd 18625  SubMndcsubmnd 18670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672
This theorem is referenced by:  zrhpsgnmhm  21137  amgmlem  26494  nn0archi  32462  amgmwlem  47849  amgmlemALT  47850
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