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Theorem subsubm 18738
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 2726 . . . . . . . 8 (Baseβ€˜π») = (Baseβ€˜π»)
21submss 18731 . . . . . . 7 (𝐴 ∈ (SubMndβ€˜π») β†’ 𝐴 βŠ† (Baseβ€˜π»))
32adantl 481 . . . . . 6 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† (Baseβ€˜π»))
4 subsubm.h . . . . . . . 8 𝐻 = (𝐺 β†Ύs 𝑆)
54submbas 18736 . . . . . . 7 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝑆 = (Baseβ€˜π»))
65adantr 480 . . . . . 6 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝑆 = (Baseβ€˜π»))
73, 6sseqtrrd 4018 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† 𝑆)
8 eqid 2726 . . . . . . 7 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
98submss 18731 . . . . . 6 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
109adantr 480 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
117, 10sstrd 3987 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† (Baseβ€˜πΊ))
12 eqid 2726 . . . . . . 7 (0gβ€˜πΊ) = (0gβ€˜πΊ)
134, 12subm0 18737 . . . . . 6 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ (0gβ€˜πΊ) = (0gβ€˜π»))
1413adantr 480 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (0gβ€˜πΊ) = (0gβ€˜π»))
15 eqid 2726 . . . . . . 7 (0gβ€˜π») = (0gβ€˜π»)
1615subm0cl 18733 . . . . . 6 (𝐴 ∈ (SubMndβ€˜π») β†’ (0gβ€˜π») ∈ 𝐴)
1716adantl 481 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (0gβ€˜π») ∈ 𝐴)
1814, 17eqeltrd 2827 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (0gβ€˜πΊ) ∈ 𝐴)
194oveq1i 7414 . . . . . . 7 (𝐻 β†Ύs 𝐴) = ((𝐺 β†Ύs 𝑆) β†Ύs 𝐴)
20 ressabs 17200 . . . . . . 7 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆) β†’ ((𝐺 β†Ύs 𝑆) β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
2119, 20eqtrid 2778 . . . . . 6 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆) β†’ (𝐻 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
227, 21syldan 590 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐻 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
23 eqid 2726 . . . . . . 7 (𝐻 β†Ύs 𝐴) = (𝐻 β†Ύs 𝐴)
2423submmnd 18735 . . . . . 6 (𝐴 ∈ (SubMndβ€˜π») β†’ (𝐻 β†Ύs 𝐴) ∈ Mnd)
2524adantl 481 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐻 β†Ύs 𝐴) ∈ Mnd)
2622, 25eqeltrrd 2828 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐺 β†Ύs 𝐴) ∈ Mnd)
27 submrcl 18724 . . . . . 6 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝐺 ∈ Mnd)
2827adantr 480 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐺 ∈ Mnd)
29 eqid 2726 . . . . . 6 (𝐺 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴)
308, 12, 29issubm2 18726 . . . . 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 1339 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 ∈ (SubMndβ€˜πΊ))
3332, 7jca 511 . 2 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆))
34 simprr 770 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐴 βŠ† 𝑆)
355adantr 480 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝑆 = (Baseβ€˜π»))
3634, 35sseqtrd 4017 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐴 βŠ† (Baseβ€˜π»))
3713adantr 480 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (0gβ€˜πΊ) = (0gβ€˜π»))
3812subm0cl 18733 . . . . 5 (𝐴 ∈ (SubMndβ€˜πΊ) β†’ (0gβ€˜πΊ) ∈ 𝐴)
3938ad2antrl 725 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (0gβ€˜πΊ) ∈ 𝐴)
4037, 39eqeltrrd 2828 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (0gβ€˜π») ∈ 𝐴)
4121adantrl 713 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐻 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
4229submmnd 18735 . . . . 5 (𝐴 ∈ (SubMndβ€˜πΊ) β†’ (𝐺 β†Ύs 𝐴) ∈ Mnd)
4342ad2antrl 725 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐺 β†Ύs 𝐴) ∈ Mnd)
4441, 43eqeltrd 2827 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐻 β†Ύs 𝐴) ∈ Mnd)
454submmnd 18735 . . . . 5 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝐻 ∈ Mnd)
4645adantr 480 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐻 ∈ Mnd)
471, 15, 23issubm2 18726 . . . 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 1339 . 2 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐴 ∈ (SubMndβ€˜π»))
5033, 49impbida 798 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150   β†Ύs cress 17179  0gc0g 17391  Mndcmnd 18664  SubMndcsubmnd 18709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  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 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-0g 17393  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-submnd 18711
This theorem is referenced by:  zrhpsgnmhm  21472  amgmlem  26872  nn0archi  32964  amgmwlem  48105  amgmlemALT  48106
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