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Theorem ismndd 17521
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 1113 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3 simpll 750 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝜑)
4 simplrl 762 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑥𝐵)
5 simplrr 763 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑦𝐵)
6 simpr 471 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑧𝐵)
7 ismndd.a . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
83, 4, 5, 6, 7syl13anc 1478 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
98ralrimiva 3115 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
102, 9jca 501 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
1110ralrimivva 3120 . . 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 6810 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
1514, 12eleq12d 2844 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
16 eqidd 2772 . . . . . . . . 9 (𝜑𝑧 = 𝑧)
1713, 14, 16oveq123d 6814 . . . . . . . 8 (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧))
18 eqidd 2772 . . . . . . . . 9 (𝜑𝑥 = 𝑥)
1913oveqd 6810 . . . . . . . . 9 (𝜑 → (𝑦 + 𝑧) = (𝑦(+g𝐺)𝑧))
2013, 18, 19oveq123d 6814 . . . . . . . 8 (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))
2117, 20eqeq12d 2786 . . . . . . 7 (𝜑 → (((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2212, 21raleqbidv 3301 . . . . . 6 (𝜑 → (∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2315, 22anbi12d 616 . . . . 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 3301 . . . 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 3301 . . 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 222 . 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 2852 . . 3 (𝜑0 ∈ (Base‘𝐺))
2912eleq2d 2836 . . . . . 6 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐺)))
3029biimpar 463 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥𝐵)
3113adantr 466 . . . . . . . 8 ((𝜑𝑥𝐵) → + = (+g𝐺))
3231oveqd 6810 . . . . . . 7 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = ( 0 (+g𝐺)𝑥))
33 ismndd.i . . . . . . 7 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
3432, 33eqtr3d 2807 . . . . . 6 ((𝜑𝑥𝐵) → ( 0 (+g𝐺)𝑥) = 𝑥)
3531oveqd 6810 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = (𝑥(+g𝐺) 0 ))
36 ismndd.j . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
3735, 36eqtr3d 2807 . . . . . 6 ((𝜑𝑥𝐵) → (𝑥(+g𝐺) 0 ) = 𝑥)
3834, 37jca 501 . . . . 5 ((𝜑𝑥𝐵) → (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
3930, 38syldan 579 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐺)) → (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
4039ralrimiva 3115 . . 3 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
41 oveq1 6800 . . . . . . 7 (𝑢 = 0 → (𝑢(+g𝐺)𝑥) = ( 0 (+g𝐺)𝑥))
4241eqeq1d 2773 . . . . . 6 (𝑢 = 0 → ((𝑢(+g𝐺)𝑥) = 𝑥 ↔ ( 0 (+g𝐺)𝑥) = 𝑥))
43 oveq2 6801 . . . . . . 7 (𝑢 = 0 → (𝑥(+g𝐺)𝑢) = (𝑥(+g𝐺) 0 ))
4443eqeq1d 2773 . . . . . 6 (𝑢 = 0 → ((𝑥(+g𝐺)𝑢) = 𝑥 ↔ (𝑥(+g𝐺) 0 ) = 𝑥))
4542, 44anbi12d 616 . . . . 5 (𝑢 = 0 → (((𝑢(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑢) = 𝑥) ↔ (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥)))
4645ralbidv 3135 . . . 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)))
4746rspcev 3460 . . 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))
4828, 40, 47syl2anc 573 . 2 (𝜑 → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑢) = 𝑥))
49 eqid 2771 . . 3 (Base‘𝐺) = (Base‘𝐺)
50 eqid 2771 . . 3 (+g𝐺) = (+g𝐺)
5149, 50ismnd 17505 . 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)))
5226, 48, 51sylanbrc 572 1 (𝜑𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  Mndcmnd 17502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-mgm 17450  df-sgrp 17492  df-mnd 17503
This theorem is referenced by:  issubmnd  17526  prdsmndd  17531  imasmnd2  17535  frmdmnd  17604  isgrpde  17651  oppgmnd  17991  isringd  18793  iscrngd  18794  xrsmcmn  19984  xrs1mnd  19999
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