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Theorem submomnd 30226
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 478 . 2 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ Mnd)
2 omndtos 30221 . . . 4 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
32adantr 473 . . 3 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → 𝑀 ∈ Toset)
4 reldmress 16251 . . . . . . . 8 Rel dom ↾s
54ovprc2 6917 . . . . . . 7 𝐴 ∈ V → (𝑀s 𝐴) = ∅)
65fveq2d 6415 . . . . . 6 𝐴 ∈ V → (Base‘(𝑀s 𝐴)) = (Base‘∅))
76adantl 474 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = (Base‘∅))
8 base0 16237 . . . . 5 ∅ = (Base‘∅)
97, 8syl6eqr 2851 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = ∅)
10 eqid 2799 . . . . . . . 8 (Base‘(𝑀s 𝐴)) = (Base‘(𝑀s 𝐴))
11 eqid 2799 . . . . . . . 8 (0g‘(𝑀s 𝐴)) = (0g‘(𝑀s 𝐴))
1210, 11mndidcl 17623 . . . . . . 7 ((𝑀s 𝐴) ∈ Mnd → (0g‘(𝑀s 𝐴)) ∈ (Base‘(𝑀s 𝐴)))
1312ne0d 4122 . . . . . 6 ((𝑀s 𝐴) ∈ Mnd → (Base‘(𝑀s 𝐴)) ≠ ∅)
1413ad2antlr 719 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) ≠ ∅)
1514neneqd 2976 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑀s 𝐴)) = ∅)
169, 15condan 853 . . 3 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → 𝐴 ∈ V)
17 resstos 30176 . . 3 ((𝑀 ∈ Toset ∧ 𝐴 ∈ V) → (𝑀s 𝐴) ∈ Toset)
183, 16, 17syl2anc 580 . 2 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ Toset)
19 simplll 792 . . . . . 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 2799 . . . . . . . . . . 11 (𝑀s 𝐴) = (𝑀s 𝐴)
21 eqid 2799 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
2220, 21ressbas 16255 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑀)) = (Base‘(𝑀s 𝐴)))
23 inss2 4029 . . . . . . . . . 10 (𝐴 ∩ (Base‘𝑀)) ⊆ (Base‘𝑀)
2422, 23syl6eqssr 3852 . . . . . . . . 9 (𝐴 ∈ V → (Base‘(𝑀s 𝐴)) ⊆ (Base‘𝑀))
2516, 24syl 17 . . . . . . . 8 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (Base‘(𝑀s 𝐴)) ⊆ (Base‘𝑀))
2625ad2antrr 718 . . . . . . 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 1276 . . . . . . 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 3799 . . . . . 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 1278 . . . . . . 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 3799 . . . . . 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 1280 . . . . . . 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 3799 . . . . . 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 2799 . . . . . . . . . . 11 (le‘𝑀) = (le‘𝑀)
3420, 33ressle 16374 . . . . . . . . . 10 (𝐴 ∈ V → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3516, 34syl 17 . . . . . . . . 9 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3635adantr 473 . . . . . . . 8 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3736breqd 4854 . . . . . . 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 470 . . . . . 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 2799 . . . . . . 7 (+g𝑀) = (+g𝑀)
4021, 33, 39omndadd 30222 . . . . . 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 1503 . . . . 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 473 . . . . . . . . 9 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → 𝐴 ∈ V)
4320, 39ressplusg 16314 . . . . . . . . 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 6895 . . . . . . 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 6895 . . . . . . 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 4857 . . . . . 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 473 . . . . 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 224 . . . 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 402 . . 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 3153 . 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 2799 . . 3 (+g‘(𝑀s 𝐴)) = (+g‘(𝑀s 𝐴))
54 eqid 2799 . . 3 (le‘(𝑀s 𝐴)) = (le‘(𝑀s 𝐴))
5510, 53, 54isomnd 30217 . 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 1444 1 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ oMnd)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  Vcvv 3385  cin 3768  wss 3769  c0 4115   class class class wbr 4843  cfv 6101  (class class class)co 6878  Basecbs 16184  s cress 16185  +gcplusg 16267  lecple 16274  0gc0g 16415  Tosetctos 17348  Mndcmnd 17609  oMndcomnd 30213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-dec 11784  df-ndx 16187  df-slot 16188  df-base 16190  df-sets 16191  df-ress 16192  df-plusg 16280  df-ple 16287  df-0g 16417  df-poset 17261  df-toset 17349  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-omnd 30215
This theorem is referenced by:  suborng  30331  nn0omnd  30357
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