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Theorem subsubm 18632
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 18625 . . . . . . 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 18630 . . . . . . 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 3986 . . . . 5 ((𝑆 ∈ (SubMndβ€˜πΊ) ∧ 𝐴 ∈ (SubMndβ€˜π»)) β†’ 𝐴 βŠ† 𝑆)
8 eqid 2733 . . . . . . 7 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
98submss 18625 . . . . . 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 3955 . . . 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 18631 . . . . . 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 18627 . . . . . 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 7368 . . . . . . 7 (𝐻 β†Ύs 𝐴) = ((𝐺 β†Ύs 𝑆) β†Ύs 𝐴)
20 ressabs 17135 . . . . . . 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 18629 . . . . . 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 18618 . . . . . 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 18620 . . . . 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 3985 . . 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 18627 . . . . 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 18629 . . . . 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 18629 . . . . 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 18620 . . . 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 3911  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088   β†Ύs cress 17117  0gc0g 17326  Mndcmnd 18561  SubMndcsubmnd 18605
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607
This theorem is referenced by:  zrhpsgnmhm  21004  amgmlem  26355  nn0archi  32186  amgmwlem  47335  amgmlemALT  47336
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