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Theorem ismndd 18583
Description: Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ismndd.b (𝜑𝐵 = (Base‘𝐺))
ismndd.p (𝜑+ = (+g𝐺))
ismndd.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
ismndd.a ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
ismndd.z (𝜑0𝐵)
ismndd.i ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
ismndd.j ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
Assertion
Ref Expression
ismndd (𝜑𝐺 ∈ Mnd)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐺,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥, 0
Allowed substitution hints:   + (𝑥,𝑦,𝑧)   0 (𝑦,𝑧)

Proof of Theorem ismndd
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ismndd.c . . . . . 6 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
213expb 1121 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3 simpll 766 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝜑)
4 simplrl 776 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑥𝐵)
5 simplrr 777 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑦𝐵)
6 simpr 486 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑧𝐵)
7 ismndd.a . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
83, 4, 5, 6, 7syl13anc 1373 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
98ralrimiva 3140 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
102, 9jca 513 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
1110ralrimivva 3194 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
12 ismndd.b . . . 4 (𝜑𝐵 = (Base‘𝐺))
13 ismndd.p . . . . . . . 8 (𝜑+ = (+g𝐺))
1413oveqd 7375 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
1514, 12eleq12d 2828 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
16 eqidd 2734 . . . . . . . . 9 (𝜑𝑧 = 𝑧)
1713, 14, 16oveq123d 7379 . . . . . . . 8 (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧))
18 eqidd 2734 . . . . . . . . 9 (𝜑𝑥 = 𝑥)
1913oveqd 7375 . . . . . . . . 9 (𝜑 → (𝑦 + 𝑧) = (𝑦(+g𝐺)𝑧))
2013, 18, 19oveq123d 7379 . . . . . . . 8 (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))
2117, 20eqeq12d 2749 . . . . . . 7 (𝜑 → (((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2212, 21raleqbidv 3318 . . . . . 6 (𝜑 → (∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2315, 22anbi12d 632 . . . . 5 (𝜑 → (((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ↔ ((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))))
2412, 23raleqbidv 3318 . . . 4 (𝜑 → (∀𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ↔ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))))
2512, 24raleqbidv 3318 . . 3 (𝜑 → (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))))
2611, 25mpbid 231 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
27 ismndd.z . . . 4 (𝜑0𝐵)
2827, 12eleqtrd 2836 . . 3 (𝜑0 ∈ (Base‘𝐺))
2912eleq2d 2820 . . . . . 6 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐺)))
3029biimpar 479 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥𝐵)
3113adantr 482 . . . . . . . 8 ((𝜑𝑥𝐵) → + = (+g𝐺))
3231oveqd 7375 . . . . . . 7 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = ( 0 (+g𝐺)𝑥))
33 ismndd.i . . . . . . 7 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
3432, 33eqtr3d 2775 . . . . . 6 ((𝜑𝑥𝐵) → ( 0 (+g𝐺)𝑥) = 𝑥)
3531oveqd 7375 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = (𝑥(+g𝐺) 0 ))
36 ismndd.j . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
3735, 36eqtr3d 2775 . . . . . 6 ((𝜑𝑥𝐵) → (𝑥(+g𝐺) 0 ) = 𝑥)
3834, 37jca 513 . . . . 5 ((𝜑𝑥𝐵) → (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
3930, 38syldan 592 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐺)) → (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
4039ralrimiva 3140 . . 3 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
41 oveq1 7365 . . . . . 6 (𝑢 = 0 → (𝑢(+g𝐺)𝑥) = ( 0 (+g𝐺)𝑥))
4241eqeq1d 2735 . . . . 5 (𝑢 = 0 → ((𝑢(+g𝐺)𝑥) = 𝑥 ↔ ( 0 (+g𝐺)𝑥) = 𝑥))
4342ovanraleqv 7382 . . . 4 (𝑢 = 0 → (∀𝑥 ∈ (Base‘𝐺)((𝑢(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑢) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥)))
4443rspcev 3580 . . 3 (( 0 ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥)) → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑢) = 𝑥))
4528, 40, 44syl2anc 585 . 2 (𝜑 → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑢) = 𝑥))
46 eqid 2733 . . 3 (Base‘𝐺) = (Base‘𝐺)
47 eqid 2733 . . 3 (+g𝐺) = (+g𝐺)
4846, 47ismnd 18564 . 2 (𝐺 ∈ Mnd ↔ (∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))) ∧ ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑢) = 𝑥)))
4926, 45, 48sylanbrc 584 1 (𝜑𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Mndcmnd 18561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-mgm 18502  df-sgrp 18551  df-mnd 18562
This theorem is referenced by:  issubmnd  18588  prdsmndd  18594  imasmnd2  18598  frmdmnd  18674  isgrpde  18776  oppgmnd  19140  isringd  20014  iscrngd  20015  xrsmcmn  20836  xrs1mnd  20851  idlsrgmnd  32304  bj-endmnd  35835  endmndlem  47121
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