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Theorem ismndd 17514
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 1114 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3 simpll 807 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝜑)
4 simplrl 819 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑥𝐵)
5 simplrr 820 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑦𝐵)
6 simpr 479 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑧𝐵)
7 ismndd.a . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
83, 4, 5, 6, 7syl13anc 1479 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
98ralrimiva 3104 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
102, 9jca 555 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
1110ralrimivva 3109 . . 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 6830 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
1514, 12eleq12d 2833 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
16 eqidd 2761 . . . . . . . . 9 (𝜑𝑧 = 𝑧)
1713, 14, 16oveq123d 6834 . . . . . . . 8 (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧))
18 eqidd 2761 . . . . . . . . 9 (𝜑𝑥 = 𝑥)
1913oveqd 6830 . . . . . . . . 9 (𝜑 → (𝑦 + 𝑧) = (𝑦(+g𝐺)𝑧))
2013, 18, 19oveq123d 6834 . . . . . . . 8 (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))
2117, 20eqeq12d 2775 . . . . . . 7 (𝜑 → (((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2212, 21raleqbidv 3291 . . . . . 6 (𝜑 → (∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2315, 22anbi12d 749 . . . . 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 3291 . . . 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 3291 . . 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 2841 . . 3 (𝜑0 ∈ (Base‘𝐺))
2912eleq2d 2825 . . . . . 6 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐺)))
3029biimpar 503 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥𝐵)
3113adantr 472 . . . . . . . 8 ((𝜑𝑥𝐵) → + = (+g𝐺))
3231oveqd 6830 . . . . . . 7 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = ( 0 (+g𝐺)𝑥))
33 ismndd.i . . . . . . 7 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
3432, 33eqtr3d 2796 . . . . . 6 ((𝜑𝑥𝐵) → ( 0 (+g𝐺)𝑥) = 𝑥)
3531oveqd 6830 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = (𝑥(+g𝐺) 0 ))
36 ismndd.j . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
3735, 36eqtr3d 2796 . . . . . 6 ((𝜑𝑥𝐵) → (𝑥(+g𝐺) 0 ) = 𝑥)
3834, 37jca 555 . . . . 5 ((𝜑𝑥𝐵) → (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
3930, 38syldan 488 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐺)) → (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
4039ralrimiva 3104 . . 3 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
41 oveq1 6820 . . . . . . 7 (𝑢 = 0 → (𝑢(+g𝐺)𝑥) = ( 0 (+g𝐺)𝑥))
4241eqeq1d 2762 . . . . . 6 (𝑢 = 0 → ((𝑢(+g𝐺)𝑥) = 𝑥 ↔ ( 0 (+g𝐺)𝑥) = 𝑥))
43 oveq2 6821 . . . . . . 7 (𝑢 = 0 → (𝑥(+g𝐺)𝑢) = (𝑥(+g𝐺) 0 ))
4443eqeq1d 2762 . . . . . 6 (𝑢 = 0 → ((𝑥(+g𝐺)𝑢) = 𝑥 ↔ (𝑥(+g𝐺) 0 ) = 𝑥))
4542, 44anbi12d 749 . . . . 5 (𝑢 = 0 → (((𝑢(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑢) = 𝑥) ↔ (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥)))
4645ralbidv 3124 . . . 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 3449 . . 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 696 . 2 (𝜑 → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑢) = 𝑥))
49 eqid 2760 . . 3 (Base‘𝐺) = (Base‘𝐺)
50 eqid 2760 . . 3 (+g𝐺) = (+g𝐺)
5149, 50ismnd 17498 . 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 701 1 (𝜑𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Mndcmnd 17495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-mgm 17443  df-sgrp 17485  df-mnd 17496
This theorem is referenced by:  issubmnd  17519  prdsmndd  17524  imasmnd2  17528  frmdmnd  17597  isgrpde  17644  oppgmnd  17984  isringd  18785  iscrngd  18786  xrsmcmn  19971  xrs1mnd  19986
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