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Theorem isomnd 31462
Description: A (left) ordered monoid is a monoid with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
isomnd.0 𝐵 = (Base‘𝑀)
isomnd.1 + = (+g𝑀)
isomnd.2 = (le‘𝑀)
Assertion
Ref Expression
isomnd (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
Distinct variable groups:   𝑎,𝑏,𝑐,𝐵   𝑀,𝑎,𝑏,𝑐
Allowed substitution hints:   + (𝑎,𝑏,𝑐)   (𝑎,𝑏,𝑐)

Proof of Theorem isomnd
Dummy variables 𝑙 𝑚 𝑝 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6827 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2 simpr 485 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑚))
3 fveq2 6812 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
43adantr 481 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → (Base‘𝑚) = (Base‘𝑀))
52, 4eqtrd 2777 . . . . . . . . . 10 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑀))
6 isomnd.0 . . . . . . . . . 10 𝐵 = (Base‘𝑀)
75, 6eqtr4di 2795 . . . . . . . . 9 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → 𝑣 = 𝐵)
8 raleq 3306 . . . . . . . . . . 11 (𝑣 = 𝐵 → (∀𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
98raleqbi1dv 3304 . . . . . . . . . 10 (𝑣 = 𝐵 → (∀𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
109raleqbi1dv 3304 . . . . . . . . 9 (𝑣 = 𝐵 → (∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
117, 10syl 17 . . . . . . . 8 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → (∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
1211anbi2d 629 . . . . . . 7 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → ((𝑚 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ (𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))))
1312sbcbidv 3785 . . . . . 6 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → ([(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))))
1413sbcbidv 3785 . . . . 5 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → ([(+g𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(+g𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))))
151, 14sbcied 3771 . . . 4 (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑣][(+g𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(+g𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))))
16 fvexd 6827 . . . . 5 (𝑚 = 𝑀 → (+g𝑚) ∈ V)
17 simpr 485 . . . . . . . . . . . . . 14 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → 𝑝 = (+g𝑚))
18 fveq2 6812 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
1918adantr 481 . . . . . . . . . . . . . 14 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → (+g𝑚) = (+g𝑀))
2017, 19eqtrd 2777 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → 𝑝 = (+g𝑀))
21 isomnd.1 . . . . . . . . . . . . 13 + = (+g𝑀)
2220, 21eqtr4di 2795 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → 𝑝 = + )
2322oveqd 7334 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → (𝑎𝑝𝑐) = (𝑎 + 𝑐))
2422oveqd 7334 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → (𝑏𝑝𝑐) = (𝑏 + 𝑐))
2523, 24breq12d 5100 . . . . . . . . . 10 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → ((𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐) ↔ (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))
2625imbi2d 340 . . . . . . . . 9 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → ((𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
2726ralbidv 3171 . . . . . . . 8 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → (∀𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
28272ralbidv 3209 . . . . . . 7 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → (∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
2928anbi2d 629 . . . . . 6 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → ((𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ (𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))))
3029sbcbidv 3785 . . . . 5 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → ([(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))))
3116, 30sbcied 3771 . . . 4 (𝑚 = 𝑀 → ([(+g𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ [(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)))))
32 fvexd 6827 . . . . . 6 (𝑚 = 𝑀 → (le‘𝑚) ∈ V)
33 simpr 485 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑚))
34 simpl 483 . . . . . . . . . . . . . 14 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → 𝑚 = 𝑀)
3534fveq2d 6816 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → (le‘𝑚) = (le‘𝑀))
3633, 35eqtrd 2777 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑀))
37 isomnd.2 . . . . . . . . . . . 12 = (le‘𝑀)
3836, 37eqtr4di 2795 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → 𝑙 = )
3938breqd 5098 . . . . . . . . . 10 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → (𝑎𝑙𝑏𝑎 𝑏))
4038breqd 5098 . . . . . . . . . 10 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → ((𝑎 + 𝑐)𝑙(𝑏 + 𝑐) ↔ (𝑎 + 𝑐) (𝑏 + 𝑐)))
4139, 40imbi12d 344 . . . . . . . . 9 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → ((𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
4241ralbidv 3171 . . . . . . . 8 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → (∀𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ ∀𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
43422ralbidv 3209 . . . . . . 7 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → (∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
4443anbi2d 629 . . . . . 6 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → ((𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))) ↔ (𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))))
4532, 44sbcied 3771 . . . . 5 (𝑚 = 𝑀 → ([(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))) ↔ (𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))))
46 eleq1 2825 . . . . . 6 (𝑚 = 𝑀 → (𝑚 ∈ Toset ↔ 𝑀 ∈ Toset))
4746anbi1d 630 . . . . 5 (𝑚 = 𝑀 → ((𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))) ↔ (𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))))
4845, 47bitrd 278 . . . 4 (𝑚 = 𝑀 → ([(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))) ↔ (𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))))
4915, 31, 483bitrd 304 . . 3 (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑣][(+g𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))) ↔ (𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))))
50 df-omnd 31460 . . 3 oMnd = {𝑚 ∈ Mnd ∣ [(Base‘𝑚) / 𝑣][(+g𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))}
5149, 50elrab2 3637 . 2 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ (𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))))
52 3anass 1094 . 2 ((𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))) ↔ (𝑀 ∈ Mnd ∧ (𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))))
5351, 52bitr4i 277 1 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3062  Vcvv 3441  [wsbc 3726   class class class wbr 5087  cfv 6466  (class class class)co 7317  Basecbs 16989  +gcplusg 17039  lecple 17046  Tosetctos 18211  Mndcmnd 18462  oMndcomnd 31458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708  ax-nul 5245
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rab 3405  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-iota 6418  df-fv 6474  df-ov 7320  df-omnd 31460
This theorem is referenced by:  omndmnd  31465  omndtos  31466  omndadd  31467  submomnd  31471  xrge0omnd  31472  reofld  31682
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