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Theorem submomnd 33087
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 33082 . . . 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 6910 . . . . . 6 𝐴 ∈ V → (Base‘(𝑀s 𝐴)) = (Base‘∅))
76adantl 481 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = (Base‘∅))
8 base0 17252 . . . . 5 ∅ = (Base‘∅)
97, 8eqtr4di 2795 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = ∅)
10 eqid 2737 . . . . . . . 8 (Base‘(𝑀s 𝐴)) = (Base‘(𝑀s 𝐴))
11 eqid 2737 . . . . . . . 8 (0g‘(𝑀s 𝐴)) = (0g‘(𝑀s 𝐴))
1210, 11mndidcl 18762 . . . . . . 7 ((𝑀s 𝐴) ∈ Mnd → (0g‘(𝑀s 𝐴)) ∈ (Base‘(𝑀s 𝐴)))
1312ne0d 4342 . . . . . 6 ((𝑀s 𝐴) ∈ Mnd → (Base‘(𝑀s 𝐴)) ≠ ∅)
1413ad2antlr 727 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) ≠ ∅)
1514neneqd 2945 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑀s 𝐴)) = ∅)
169, 15condan 818 . . 3 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → 𝐴 ∈ V)
17 resstos 32957 . . 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 2737 . . . . . . . . . . 11 (𝑀s 𝐴) = (𝑀s 𝐴)
21 eqid 2737 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
2220, 21ressbas 17280 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑀)) = (Base‘(𝑀s 𝐴)))
23 inss2 4238 . . . . . . . . . 10 (𝐴 ∩ (Base‘𝑀)) ⊆ (Base‘𝑀)
2422, 23eqsstrrdi 4029 . . . . . . . . 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 1216 . . . . . . 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 3984 . . . . . 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 1217 . . . . . . 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 3984 . . . . . 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 1218 . . . . . . 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 3984 . . . . . 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 2737 . . . . . . . . . . 11 (le‘𝑀) = (le‘𝑀)
3420, 33ressle 17424 . . . . . . . . . 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 5154 . . . . . . 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 2737 . . . . . . 7 (+g𝑀) = (+g𝑀)
4021, 33, 39omndadd 33083 . . . . . 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 1385 . . . . 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 17334 . . . . . . . . 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 5157 . . . . . 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 3205 . 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 2737 . . 3 (+g‘(𝑀s 𝐴)) = (+g‘(𝑀s 𝐴))
54 eqid 2737 . . 3 (le‘(𝑀s 𝐴)) = (le‘(𝑀s 𝐴))
5510, 53, 54isomnd 33078 . 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 1344 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 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cin 3950  wss 3951  c0 4333   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  +gcplusg 17297  lecple 17304  0gc0g 17484  Tosetctos 18461  Mndcmnd 18747  oMndcomnd 33074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-dec 12734  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-ple 17317  df-0g 17486  df-poset 18359  df-toset 18462  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-omnd 33076
This theorem is referenced by:  suborng  33345  nn0omnd  33373
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