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Theorem submomnd 29041
 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 476 . 2 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ Mnd)
2 omndtos 29036 . . . 4 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
32adantr 480 . . 3 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → 𝑀 ∈ Toset)
4 reldmress 15753 . . . . . . . 8 Rel dom ↾s
54ovprc2 6583 . . . . . . 7 𝐴 ∈ V → (𝑀s 𝐴) = ∅)
65fveq2d 6107 . . . . . 6 𝐴 ∈ V → (Base‘(𝑀s 𝐴)) = (Base‘∅))
76adantl 481 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = (Base‘∅))
8 base0 15740 . . . . 5 ∅ = (Base‘∅)
97, 8syl6eqr 2662 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) = ∅)
10 eqid 2610 . . . . . . . 8 (Base‘(𝑀s 𝐴)) = (Base‘(𝑀s 𝐴))
11 eqid 2610 . . . . . . . 8 (0g‘(𝑀s 𝐴)) = (0g‘(𝑀s 𝐴))
1210, 11mndidcl 17131 . . . . . . 7 ((𝑀s 𝐴) ∈ Mnd → (0g‘(𝑀s 𝐴)) ∈ (Base‘(𝑀s 𝐴)))
13 ne0i 3880 . . . . . . 7 ((0g‘(𝑀s 𝐴)) ∈ (Base‘(𝑀s 𝐴)) → (Base‘(𝑀s 𝐴)) ≠ ∅)
1412, 13syl 17 . . . . . 6 ((𝑀s 𝐴) ∈ Mnd → (Base‘(𝑀s 𝐴)) ≠ ∅)
1514ad2antlr 759 . . . . 5 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → (Base‘(𝑀s 𝐴)) ≠ ∅)
1615neneqd 2787 . . . 4 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ ¬ 𝐴 ∈ V) → ¬ (Base‘(𝑀s 𝐴)) = ∅)
179, 16condan 831 . . 3 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → 𝐴 ∈ V)
18 resstos 28991 . . 3 ((𝑀 ∈ Toset ∧ 𝐴 ∈ V) → (𝑀s 𝐴) ∈ Toset)
193, 17, 18syl2anc 691 . 2 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ Toset)
20 simplll 794 . . . . . 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)
21 eqid 2610 . . . . . . . . . . 11 (𝑀s 𝐴) = (𝑀s 𝐴)
22 eqid 2610 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
2321, 22ressbas 15757 . . . . . . . . . 10 (𝐴 ∈ V → (𝐴 ∩ (Base‘𝑀)) = (Base‘(𝑀s 𝐴)))
24 inss2 3796 . . . . . . . . . 10 (𝐴 ∩ (Base‘𝑀)) ⊆ (Base‘𝑀)
2523, 24syl6eqssr 3619 . . . . . . . . 9 (𝐴 ∈ V → (Base‘(𝑀s 𝐴)) ⊆ (Base‘𝑀))
2617, 25syl 17 . . . . . . . 8 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (Base‘(𝑀s 𝐴)) ⊆ (Base‘𝑀))
2726ad2antrr 758 . . . . . . 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))
28 simplr1 1096 . . . . . . 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)))
2927, 28sseldd 3569 . . . . . 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))
30 simplr2 1097 . . . . . . 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)))
3127, 30sseldd 3569 . . . . . 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))
32 simplr3 1098 . . . . . . 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)))
3327, 32sseldd 3569 . . . . . 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))
34 eqid 2610 . . . . . . . . . . 11 (le‘𝑀) = (le‘𝑀)
3521, 34ressle 15882 . . . . . . . . . 10 (𝐴 ∈ V → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3617, 35syl 17 . . . . . . . . 9 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3736adantr 480 . . . . . . . 8 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
3837breqd 4594 . . . . . . 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))
3938biimpar 501 . . . . . 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)
40 eqid 2610 . . . . . . 7 (+g𝑀) = (+g𝑀)
4122, 34, 40omndadd 29037 . . . . . 6 ((𝑀 ∈ oMnd ∧ (𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀) ∧ 𝑐 ∈ (Base‘𝑀)) ∧ 𝑎(le‘𝑀)𝑏) → (𝑎(+g𝑀)𝑐)(le‘𝑀)(𝑏(+g𝑀)𝑐))
4220, 29, 31, 33, 39, 41syl131anc 1331 . . . . 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))
4317adantr 480 . . . . . . . . 9 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → 𝐴 ∈ V)
4421, 40ressplusg 15818 . . . . . . . . 9 (𝐴 ∈ V → (+g𝑀) = (+g‘(𝑀s 𝐴)))
4543, 44syl 17 . . . . . . . 8 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (+g𝑀) = (+g‘(𝑀s 𝐴)))
4645oveqd 6566 . . . . . . 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))
4743, 35syl 17 . . . . . . 7 (((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) ∧ (𝑎 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑏 ∈ (Base‘(𝑀s 𝐴)) ∧ 𝑐 ∈ (Base‘(𝑀s 𝐴)))) → (le‘𝑀) = (le‘(𝑀s 𝐴)))
4845oveqd 6566 . . . . . . 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))
4946, 47, 48breq123d 4597 . . . . . 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)))
5049adantr 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)))
5142, 50mpbid 221 . . . 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))
5251ex 449 . . 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)))
5352ralrimivvva 2955 . 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)))
54 eqid 2610 . . 3 (+g‘(𝑀s 𝐴)) = (+g‘(𝑀s 𝐴))
55 eqid 2610 . . 3 (le‘(𝑀s 𝐴)) = (le‘(𝑀s 𝐴))
5610, 54, 55isomnd 29032 . 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))))
571, 19, 53, 56syl3anbrc 1239 1 ((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ oMnd)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  lecple 15775  0gc0g 15923  Tosetctos 16856  Mndcmnd 17117  oMndcomnd 29028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-ple 15788  df-0g 15925  df-poset 16769  df-toset 16857  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-omnd 29030 This theorem is referenced by:  suborng  29146  nn0omnd  29172
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