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Theorem isomnd 31909
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 6857 . . . . 5 (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2 simpr 485 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑚))
3 fveq2 6842 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
43adantr 481 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → (Base‘𝑚) = (Base‘𝑀))
52, 4eqtrd 2776 . . . . . . . . . 10 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → 𝑣 = (Base‘𝑀))
6 isomnd.0 . . . . . . . . . 10 𝐵 = (Base‘𝑀)
75, 6eqtr4di 2794 . . . . . . . . 9 ((𝑚 = 𝑀𝑣 = (Base‘𝑚)) → 𝑣 = 𝐵)
8 raleq 3309 . . . . . . . . . . 11 (𝑣 = 𝐵 → (∀𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
98raleqbi1dv 3307 . . . . . . . . . 10 (𝑣 = 𝐵 → (∀𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑏𝐵𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐))))
109raleqbi1dv 3307 . . . . . . . . 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 3798 . . . . . 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 3798 . . . . 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 3784 . . . 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 6857 . . . . 5 (𝑚 = 𝑀 → (+g𝑚) ∈ V)
17 simpr 485 . . . . . . . . . . . . . 14 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → 𝑝 = (+g𝑚))
18 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
1918adantr 481 . . . . . . . . . . . . . 14 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → (+g𝑚) = (+g𝑀))
2017, 19eqtrd 2776 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → 𝑝 = (+g𝑀))
21 isomnd.1 . . . . . . . . . . . . 13 + = (+g𝑀)
2220, 21eqtr4di 2794 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → 𝑝 = + )
2322oveqd 7374 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → (𝑎𝑝𝑐) = (𝑎 + 𝑐))
2422oveqd 7374 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → (𝑏𝑝𝑐) = (𝑏 + 𝑐))
2523, 24breq12d 5118 . . . . . . . . . 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 3174 . . . . . . . 8 ((𝑚 = 𝑀𝑝 = (+g𝑚)) → (∀𝑐𝐵 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)) ↔ ∀𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐))))
28272ralbidv 3212 . . . . . . 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 3798 . . . . 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 3784 . . . 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 6857 . . . . . 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 6846 . . . . . . . . . . . . 13 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → (le‘𝑚) = (le‘𝑀))
3633, 35eqtrd 2776 . . . . . . . . . . . 12 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → 𝑙 = (le‘𝑀))
37 isomnd.2 . . . . . . . . . . . 12 = (le‘𝑀)
3836, 37eqtr4di 2794 . . . . . . . . . . 11 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → 𝑙 = )
3938breqd 5116 . . . . . . . . . 10 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → (𝑎𝑙𝑏𝑎 𝑏))
4038breqd 5116 . . . . . . . . . 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 3174 . . . . . . . 8 ((𝑚 = 𝑀𝑙 = (le‘𝑚)) → (∀𝑐𝐵 (𝑎𝑙𝑏 → (𝑎 + 𝑐)𝑙(𝑏 + 𝑐)) ↔ ∀𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
43422ralbidv 3212 . . . . . . 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 3784 . . . . 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 31907 . . 3 oMnd = {𝑚 ∈ Mnd ∣ [(Base‘𝑚) / 𝑣][(+g𝑚) / 𝑝][(le‘𝑚) / 𝑙](𝑚 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))}
5149, 50elrab2 3648 . 2 (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ (𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐)))))
52 3anass 1095 . 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 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  [wsbc 3739   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  lecple 17140  Tosetctos 18305  Mndcmnd 18556  oMndcomnd 31905
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360  df-omnd 31907
This theorem is referenced by:  omndmnd  31912  omndtos  31913  omndadd  31914  submomnd  31918  xrge0omnd  31919  reofld  32136
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