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Theorem subsubm 18775
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 2728 . . . . . . . 8 (Baseβ€˜π») = (Baseβ€˜π»)
21submss 18768 . . . . . . 7 (𝐴 ∈ (SubMndβ€˜π») β†’ 𝐴 βŠ† (Baseβ€˜π»))
32adantl 480 . . . . . 6 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† (Baseβ€˜π»))
4 subsubm.h . . . . . . . 8 𝐻 = (𝐺 β†Ύs 𝑆)
54submbas 18773 . . . . . . 7 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝑆 = (Baseβ€˜π»))
65adantr 479 . . . . . 6 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝑆 = (Baseβ€˜π»))
73, 6sseqtrrd 4023 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† 𝑆)
8 eqid 2728 . . . . . . 7 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
98submss 18768 . . . . . 6 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
109adantr 479 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝑆 βŠ† (Baseβ€˜πΊ))
117, 10sstrd 3992 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† (Baseβ€˜πΊ))
12 eqid 2728 . . . . . . 7 (0gβ€˜πΊ) = (0gβ€˜πΊ)
134, 12subm0 18774 . . . . . 6 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ (0gβ€˜πΊ) = (0gβ€˜π»))
1413adantr 479 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (0gβ€˜πΊ) = (0gβ€˜π»))
15 eqid 2728 . . . . . . 7 (0gβ€˜π») = (0gβ€˜π»)
1615subm0cl 18770 . . . . . 6 (𝐴 ∈ (SubMndβ€˜π») β†’ (0gβ€˜π») ∈ 𝐴)
1716adantl 480 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (0gβ€˜π») ∈ 𝐴)
1814, 17eqeltrd 2829 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (0gβ€˜πΊ) ∈ 𝐴)
194oveq1i 7436 . . . . . . 7 (𝐻 β†Ύs 𝐴) = ((𝐺 β†Ύs 𝑆) β†Ύs 𝐴)
20 ressabs 17237 . . . . . . 7 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆) β†’ ((𝐺 β†Ύs 𝑆) β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
2119, 20eqtrid 2780 . . . . . 6 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆) β†’ (𝐻 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
227, 21syldan 589 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐻 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
23 eqid 2728 . . . . . . 7 (𝐻 β†Ύs 𝐴) = (𝐻 β†Ύs 𝐴)
2423submmnd 18772 . . . . . 6 (𝐴 ∈ (SubMndβ€˜π») β†’ (𝐻 β†Ύs 𝐴) ∈ Mnd)
2524adantl 480 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐻 β†Ύs 𝐴) ∈ Mnd)
2622, 25eqeltrrd 2830 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐺 β†Ύs 𝐴) ∈ Mnd)
27 submrcl 18761 . . . . . 6 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝐺 ∈ Mnd)
2827adantr 479 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐺 ∈ Mnd)
29 eqid 2728 . . . . . 6 (𝐺 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴)
308, 12, 29issubm2 18763 . . . . 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 510 . 2 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆))
34 simprr 771 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐴 βŠ† 𝑆)
355adantr 479 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝑆 = (Baseβ€˜π»))
3634, 35sseqtrd 4022 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐴 βŠ† (Baseβ€˜π»))
3713adantr 479 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (0gβ€˜πΊ) = (0gβ€˜π»))
3812subm0cl 18770 . . . . 5 (𝐴 ∈ (SubMndβ€˜πΊ) β†’ (0gβ€˜πΊ) ∈ 𝐴)
3938ad2antrl 726 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (0gβ€˜πΊ) ∈ 𝐴)
4037, 39eqeltrrd 2830 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (0gβ€˜π») ∈ 𝐴)
4121adantrl 714 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐻 β†Ύs 𝐴) = (𝐺 β†Ύs 𝐴))
4229submmnd 18772 . . . . 5 (𝐴 ∈ (SubMndβ€˜πΊ) β†’ (𝐺 β†Ύs 𝐴) ∈ Mnd)
4342ad2antrl 726 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐺 β†Ύs 𝐴) ∈ Mnd)
4441, 43eqeltrd 2829 . . 3 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ (𝐻 β†Ύs 𝐴) ∈ Mnd)
454submmnd 18772 . . . . 5 (𝑆 ∈ (SubMndβ€˜πΊ) β†’ 𝐻 ∈ Mnd)
4645adantr 479 . . . 4 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ (𝐴 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 βŠ† 𝑆)) β†’ 𝐻 ∈ Mnd)
471, 15, 23issubm2 18763 . . . 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 799 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187   β†Ύs cress 17216  0gc0g 17428  Mndcmnd 18701  SubMndcsubmnd 18746
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748
This theorem is referenced by:  zrhpsgnmhm  21523  amgmlem  26942  nn0archi  33083  amgmwlem  48313  amgmlemALT  48314
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