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Theorem ismndd 18723
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 1117 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3 simpll 765 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝜑)
4 simplrl 775 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑥𝐵)
5 simplrr 776 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑦𝐵)
6 simpr 483 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑧𝐵)
7 ismndd.a . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
83, 4, 5, 6, 7syl13anc 1369 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
98ralrimiva 3143 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
102, 9jca 510 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
1110ralrimivva 3198 . . 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 7443 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
1514, 12eleq12d 2823 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
16 eqidd 2729 . . . . . . . . 9 (𝜑𝑧 = 𝑧)
1713, 14, 16oveq123d 7447 . . . . . . . 8 (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧))
18 eqidd 2729 . . . . . . . . 9 (𝜑𝑥 = 𝑥)
1913oveqd 7443 . . . . . . . . 9 (𝜑 → (𝑦 + 𝑧) = (𝑦(+g𝐺)𝑧))
2013, 18, 19oveq123d 7447 . . . . . . . 8 (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))
2117, 20eqeq12d 2744 . . . . . . 7 (𝜑 → (((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2212, 21raleqbidv 3340 . . . . . 6 (𝜑 → (∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2315, 22anbi12d 630 . . . . 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 3340 . . . 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 3340 . . 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 2831 . . 3 (𝜑0 ∈ (Base‘𝐺))
2912eleq2d 2815 . . . . . 6 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐺)))
3029biimpar 476 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐺)) → 𝑥𝐵)
3113adantr 479 . . . . . . . 8 ((𝜑𝑥𝐵) → + = (+g𝐺))
3231oveqd 7443 . . . . . . 7 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = ( 0 (+g𝐺)𝑥))
33 ismndd.i . . . . . . 7 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
3432, 33eqtr3d 2770 . . . . . 6 ((𝜑𝑥𝐵) → ( 0 (+g𝐺)𝑥) = 𝑥)
3531oveqd 7443 . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = (𝑥(+g𝐺) 0 ))
36 ismndd.j . . . . . . 7 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
3735, 36eqtr3d 2770 . . . . . 6 ((𝜑𝑥𝐵) → (𝑥(+g𝐺) 0 ) = 𝑥)
3834, 37jca 510 . . . . 5 ((𝜑𝑥𝐵) → (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
3930, 38syldan 589 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐺)) → (( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
4039ralrimiva 3143 . . 3 (𝜑 → ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺) 0 ) = 𝑥))
41 oveq1 7433 . . . . . 6 (𝑢 = 0 → (𝑢(+g𝐺)𝑥) = ( 0 (+g𝐺)𝑥))
4241eqeq1d 2730 . . . . 5 (𝑢 = 0 → ((𝑢(+g𝐺)𝑥) = 𝑥 ↔ ( 0 (+g𝐺)𝑥) = 𝑥))
4342ovanraleqv 7450 . . . 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 3611 . . 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 582 . 2 (𝜑 → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑢) = 𝑥))
46 eqid 2728 . . 3 (Base‘𝐺) = (Base‘𝐺)
47 eqid 2728 . . 3 (+g𝐺) = (+g𝐺)
4846, 47ismnd 18704 . 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 581 1 (𝜑𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  wrex 3067  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  Mndcmnd 18701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-mgm 18607  df-sgrp 18686  df-mnd 18702
This theorem is referenced by:  issubmnd  18728  prdsmndd  18734  imasmnd2  18738  frmdmnd  18818  isgrpde  18921  oppgmnd  19315  isringd  20234  iscrngd  20235  xrsmcmn  21326  xrs1mnd  21344  rloccring  33009  idlsrgmnd  33250  bj-endmnd  36830  endmndlem  48099
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