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Theorem submomnd 30718
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 488 . 2 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ Mnd)
2 omndtos 30713 . . . 4 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
32adantr 484 . . 3 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → 𝑀 ∈ Toset)
4 reldmress 16528 . . . . . . . 8 Rel dom ↾s
54ovprc2 7170 . . . . . . 7 𝐴 ∈ V → (𝑀s 𝐴) = ∅)
65fveq2d 6647 . . . . . 6 𝐴 ∈ V → (Base‘(𝑀s 𝐴)) = (Base‘∅))
76adantl 485 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = (Base‘∅))
8 base0 16514 . . . . 5 ∅ = (Base‘∅)
97, 8syl6eqr 2874 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = ∅)
10 eqid 2821 . . . . . . . 8 (Base‘(𝑀s 𝐴)) = (Base‘(𝑀s 𝐴))
11 eqid 2821 . . . . . . . 8 (0g‘(𝑀s 𝐴)) = (0g‘(𝑀s 𝐴))
1210, 11mndidcl 17904 . . . . . . 7 ((𝑀s 𝐴) ∈ Mnd → (0g‘(𝑀s 𝐴)) ∈ (Base‘(𝑀s 𝐴)))
1312ne0d 4274 . . . . . 6 ((𝑀s 𝐴) ∈ Mnd → (Base‘(𝑀s 𝐴)) ≠ ∅)
1413ad2antlr 726 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) ≠ ∅)
1514neneqd 3012 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑀s 𝐴)) = ∅)
169, 15condan 817 . . 3 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → 𝐴 ∈ V)
17 resstos 30633 . . 3 ((𝑀 ∈ Toset ∧ 𝐴 ∈ V) → (𝑀s 𝐴) ∈ Toset)
183, 16, 17syl2anc 587 . 2 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ Toset)
19 simplll 774 . . . . . 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 2821 . . . . . . . . . . 11 (𝑀s 𝐴) = (𝑀s 𝐴)
21 eqid 2821 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
2220, 21ressbas 16532 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑀)) = (Base‘(𝑀s 𝐴)))
23 inss2 4181 . . . . . . . . . 10 (𝐴 ∩ (Base‘𝑀)) ⊆ (Base‘𝑀)
2422, 23eqsstrrdi 3998 . . . . . . . . 9 (𝐴 ∈ V → (Base‘(𝑀s 𝐴)) ⊆ (Base‘𝑀))
2516, 24syl 17 . . . . . . . 8 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (Base‘(𝑀s 𝐴)) ⊆ (Base‘𝑀))
2625ad2antrr 725 . . . . . . 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 1212 . . . . . . 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 3944 . . . . . 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 1213 . . . . . . 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 3944 . . . . . 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 1214 . . . . . . 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 3944 . . . . . 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 2821 . . . . . . . . . . 11 (le‘𝑀) = (le‘𝑀)
3420, 33ressle 16650 . . . . . . . . . 10 (𝐴 ∈ V → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3516, 34syl 17 . . . . . . . . 9 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3635adantr 484 . . . . . . . 8 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3736breqd 5050 . . . . . . 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 481 . . . . . 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 2821 . . . . . . 7 (+g𝑀) = (+g𝑀)
4021, 33, 39omndadd 30714 . . . . . 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 1380 . . . . 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 484 . . . . . . . . 9 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → 𝐴 ∈ V)
4320, 39ressplusg 16590 . . . . . . . . 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 7147 . . . . . . 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 7147 . . . . . . 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 5053 . . . . . 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 484 . . . . 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 235 . . . 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 416 . . 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 3180 . 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 2821 . . 3 (+g‘(𝑀s 𝐴)) = (+g‘(𝑀s 𝐴))
54 eqid 2821 . . 3 (le‘(𝑀s 𝐴)) = (le‘(𝑀s 𝐴))
5510, 53, 54isomnd 30709 . 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 1340 1 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ oMnd)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3007  wral 3126  Vcvv 3471  cin 3909  wss 3910  c0 4266   class class class wbr 5039  cfv 6328  (class class class)co 7130  Basecbs 16461  s cress 16462  +gcplusg 16543  lecple 16550  0gc0g 16691  Tosetctos 17621  Mndcmnd 17889  oMndcomnd 30705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-dec 12077  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-ple 16563  df-0g 16693  df-poset 17534  df-toset 17622  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-omnd 30707
This theorem is referenced by:  suborng  30895  nn0omnd  30921
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