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Theorem submomnd 33070
Description: A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
submomnd ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ oMnd)

Proof of Theorem submomnd
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 484 . 2 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ Mnd)
2 omndtos 33065 . . . 4 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
32adantr 480 . . 3 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → 𝑀 ∈ Toset)
4 reldmress 17276 . . . . . . . 8 Rel dom ↾s
54ovprc2 7471 . . . . . . 7 𝐴 ∈ V → (𝑀s 𝐴) = ∅)
65fveq2d 6911 . . . . . 6 𝐴 ∈ V → (Base‘(𝑀s 𝐴)) = (Base‘∅))
76adantl 481 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = (Base‘∅))
8 base0 17250 . . . . 5 ∅ = (Base‘∅)
97, 8eqtr4di 2793 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = ∅)
10 eqid 2735 . . . . . . . 8 (Base‘(𝑀s 𝐴)) = (Base‘(𝑀s 𝐴))
11 eqid 2735 . . . . . . . 8 (0g‘(𝑀s 𝐴)) = (0g‘(𝑀s 𝐴))
1210, 11mndidcl 18775 . . . . . . 7 ((𝑀s 𝐴) ∈ Mnd → (0g‘(𝑀s 𝐴)) ∈ (Base‘(𝑀s 𝐴)))
1312ne0d 4348 . . . . . 6 ((𝑀s 𝐴) ∈ Mnd → (Base‘(𝑀s 𝐴)) ≠ ∅)
1413ad2antlr 727 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) ≠ ∅)
1514neneqd 2943 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑀s 𝐴)) = ∅)
169, 15condan 818 . . 3 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → 𝐴 ∈ V)
17 resstos 32942 . . 3 ((𝑀 ∈ Toset ∧ 𝐴 ∈ V) → (𝑀s 𝐴) ∈ Toset)
183, 16, 17syl2anc 584 . 2 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ Toset)
19 simplll 775 . . . . . 6 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → 𝑀 ∈ oMnd)
20 eqid 2735 . . . . . . . . . . 11 (𝑀s 𝐴) = (𝑀s 𝐴)
21 eqid 2735 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
2220, 21ressbas 17280 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑀)) = (Base‘(𝑀s 𝐴)))
23 inss2 4246 . . . . . . . . . 10 (𝐴 ∩ (Base‘𝑀)) ⊆ (Base‘𝑀)
2422, 23eqsstrrdi 4051 . . . . . . . . 9 (𝐴 ∈ V → (Base‘(𝑀s 𝐴)) ⊆ (Base‘𝑀))
2516, 24syl 17 . . . . . . . 8 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (Base‘(𝑀s 𝐴)) ⊆ (Base‘𝑀))
2625ad2antrr 726 . . . . . . 7 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → (Base‘(𝑀s 𝐴)) ⊆ (Base‘𝑀))
27 simplr1 1214 . . . . . . 7 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → 𝑎 ∈ (Base‘(𝑀s 𝐴)))
2826, 27sseldd 3996 . . . . . 6 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → 𝑎 ∈ (Base‘𝑀))
29 simplr2 1215 . . . . . . 7 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → 𝑏 ∈ (Base‘(𝑀s 𝐴)))
3026, 29sseldd 3996 . . . . . 6 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → 𝑏 ∈ (Base‘𝑀))
31 simplr3 1216 . . . . . . 7 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → 𝑐 ∈ (Base‘(𝑀s 𝐴)))
3226, 31sseldd 3996 . . . . . 6 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → 𝑐 ∈ (Base‘𝑀))
33 eqid 2735 . . . . . . . . . . 11 (le‘𝑀) = (le‘𝑀)
3420, 33ressle 17426 . . . . . . . . . 10 (𝐴 ∈ V → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3516, 34syl 17 . . . . . . . . 9 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3635adantr 480 . . . . . . . 8 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3736breqd 5159 . . . . . . 7 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (𝑎(le‘𝑀)𝑏𝑎(le‘(𝑀s 𝐴))𝑏))
3837biimpar 477 . . . . . 6 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → 𝑎(le‘𝑀)𝑏)
39 eqid 2735 . . . . . . 7 (+g𝑀) = (+g𝑀)
4021, 33, 39omndadd 33066 . . . . . 6 ((𝑀 ∈ oMnd ∧ (𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀) ∧ 𝑐 ∈ (Base‘𝑀)) ∧ 𝑎(le‘𝑀)𝑏) → (𝑎(+g𝑀)𝑐)(le‘𝑀)(𝑏(+g𝑀)𝑐))
4119, 28, 30, 32, 38, 40syl131anc 1382 . . . . 5 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → (𝑎(+g𝑀)𝑐)(le‘𝑀)(𝑏(+g𝑀)𝑐))
4216adantr 480 . . . . . . . . 9 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → 𝐴 ∈ V)
4320, 39ressplusg 17336 . . . . . . . . 9 (𝐴 ∈ V → (+g𝑀) = (+g‘(𝑀s 𝐴)))
4442, 43syl 17 . . . . . . . 8 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (+g𝑀) = (+g‘(𝑀s 𝐴)))
4544oveqd 7448 . . . . . . 7 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (𝑎(+g𝑀)𝑐) = (𝑎(+g‘(𝑀s 𝐴))𝑐))
4642, 34syl 17 . . . . . . 7 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
4744oveqd 7448 . . . . . . 7 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (𝑏(+g𝑀)𝑐) = (𝑏(+g‘(𝑀s 𝐴))𝑐))
4845, 46, 47breq123d 5162 . . . . . 6 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → ((𝑎(+g𝑀)𝑐)(le‘𝑀)(𝑏(+g𝑀)𝑐) ↔ (𝑎(+g‘(𝑀s 𝐴))𝑐)(le‘(𝑀s 𝐴))(𝑏(+g‘(𝑀s 𝐴))𝑐)))
4948adantr 480 . . . . 5 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → ((𝑎(+g𝑀)𝑐)(le‘𝑀)(𝑏(+g𝑀)𝑐) ↔ (𝑎(+g‘(𝑀s 𝐴))𝑐)(le‘(𝑀s 𝐴))(𝑏(+g‘(𝑀s 𝐴))𝑐)))
5041, 49mpbid 232 . . . 4 ((((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) ∧ 𝑎(le‘(𝑀s 𝐴))𝑏) → (𝑎(+g‘(𝑀s 𝐴))𝑐)(le‘(𝑀s 𝐴))(𝑏(+g‘(𝑀s 𝐴))𝑐))
5150ex 412 . . 3 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (𝑎(le‘(𝑀s 𝐴))𝑏 → (𝑎(+g‘(𝑀s 𝐴))𝑐)(le‘(𝑀s 𝐴))(𝑏(+g‘(𝑀s 𝐴))𝑐)))
5251ralrimivvva 3203 . 2 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → ∀𝑎 ∈ (Base‘(𝑀s 𝐴))∀𝑏 ∈ (Base‘(𝑀s 𝐴))∀𝑐 ∈ (Base‘(𝑀s 𝐴))(𝑎(le‘(𝑀s 𝐴))𝑏 → (𝑎(+g‘(𝑀s 𝐴))𝑐)(le‘(𝑀s 𝐴))(𝑏(+g‘(𝑀s 𝐴))𝑐)))
53 eqid 2735 . . 3 (+g‘(𝑀s 𝐴)) = (+g‘(𝑀s 𝐴))
54 eqid 2735 . . 3 (le‘(𝑀s 𝐴)) = (le‘(𝑀s 𝐴))
5510, 53, 54isomnd 33061 . 2 ((𝑀s 𝐴) ∈ oMnd ↔ ((𝑀s 𝐴) ∈ Mnd ∧ (𝑀s 𝐴) ∈ Toset ∧ ∀𝑎 ∈ (Base‘(𝑀s 𝐴))∀𝑏 ∈ (Base‘(𝑀s 𝐴))∀𝑐 ∈ (Base‘(𝑀s 𝐴))(𝑎(le‘(𝑀s 𝐴))𝑏 → (𝑎(+g‘(𝑀s 𝐴))𝑐)(le‘(𝑀s 𝐴))(𝑏(+g‘(𝑀s 𝐴))𝑐))))
561, 18, 52, 55syl3anbrc 1342 1 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ oMnd)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  Vcvv 3478  cin 3962  wss 3963  c0 4339   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  +gcplusg 17298  lecple 17305  0gc0g 17486  Tosetctos 18474  Mndcmnd 18760  oMndcomnd 33057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-dec 12732  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-ple 17318  df-0g 17488  df-poset 18371  df-toset 18475  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-omnd 33059
This theorem is referenced by:  suborng  33325  nn0omnd  33353
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