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Theorem isnsgrp 18360
Description: A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.)
Hypotheses
Ref Expression
issgrpn0.b 𝐵 = (Base‘𝑀)
issgrpn0.o = (+g𝑀)
Assertion
Ref Expression
isnsgrp ((𝑋𝐵𝑌𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → 𝑀 ∉ Smgrp))

Proof of Theorem isnsgrp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1189 . . . . . . 7 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑋𝐵)
2 oveq1 7275 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
32oveq1d 7283 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ((𝑥 𝑦) 𝑧) = ((𝑋 𝑦) 𝑧))
4 oveq1 7275 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
53, 4eqeq12d 2755 . . . . . . . . . . 11 (𝑥 = 𝑋 → (((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
65notbid 317 . . . . . . . . . 10 (𝑥 = 𝑋 → (¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
76rexbidv 3227 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
87rexbidv 3227 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑦𝐵𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
98adantl 481 . . . . . . 7 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑥 = 𝑋) → (∃𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑦𝐵𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
10 simpl2 1190 . . . . . . . 8 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑌𝐵)
11 oveq2 7276 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
1211oveq1d 7283 . . . . . . . . . . . 12 (𝑦 = 𝑌 → ((𝑋 𝑦) 𝑧) = ((𝑋 𝑌) 𝑧))
13 oveq1 7275 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1413oveq2d 7284 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
1512, 14eqeq12d 2755 . . . . . . . . . . 11 (𝑦 = 𝑌 → (((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1615notbid 317 . . . . . . . . . 10 (𝑦 = 𝑌 → (¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1716adantl 481 . . . . . . . . 9 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑦 = 𝑌) → (¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1817rexbidv 3227 . . . . . . . 8 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑦 = 𝑌) → (∃𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ∃𝑧𝐵 ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
19 simpl3 1191 . . . . . . . . 9 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑍𝐵)
20 oveq2 7276 . . . . . . . . . . . 12 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧) = ((𝑋 𝑌) 𝑍))
21 oveq2 7276 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
2221oveq2d 7284 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
2320, 22eqeq12d 2755 . . . . . . . . . . 11 (𝑧 = 𝑍 → (((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2423notbid 317 . . . . . . . . . 10 (𝑧 = 𝑍 → (¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2524adantl 481 . . . . . . . . 9 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑧 = 𝑍) → (¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
26 neneq 2950 . . . . . . . . . 10 (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
2726adantl 481 . . . . . . . . 9 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
2819, 25, 27rspcedvd 3563 . . . . . . . 8 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑧𝐵 ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)))
2910, 18, 28rspcedvd 3563 . . . . . . 7 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑦𝐵𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)))
301, 9, 29rspcedvd 3563 . . . . . 6 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
31 rexnal 3167 . . . . . . . 8 (∃𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
32312rexbii 3180 . . . . . . 7 (∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑥𝐵𝑦𝐵 ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
33 rexnal2 3188 . . . . . . 7 (∃𝑥𝐵𝑦𝐵 ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3432, 33bitr2i 275 . . . . . 6 (¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3530, 34sylibr 233 . . . . 5 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3635intnand 488 . . . 4 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
37 issgrpn0.b . . . . 5 𝐵 = (Base‘𝑀)
38 issgrpn0.o . . . . 5 = (+g𝑀)
3937, 38issgrp 18357 . . . 4 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
4036, 39sylnibr 328 . . 3 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ 𝑀 ∈ Smgrp)
41 df-nel 3051 . . 3 (𝑀 ∉ Smgrp ↔ ¬ 𝑀 ∈ Smgrp)
4240, 41sylibr 233 . 2 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑀 ∉ Smgrp)
4342ex 412 1 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → 𝑀 ∉ Smgrp))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1085   = wceq 1541  wcel 2109  wne 2944  wnel 3050  wral 3065  wrex 3066  cfv 6430  (class class class)co 7268  Basecbs 16893  +gcplusg 16943  Mgmcmgm 18305  Smgrpcsgrp 18355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-nul 5233
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-ov 7271  df-sgrp 18356
This theorem is referenced by:  mgm2nsgrplem4  18541  xrsnsgrp  20615
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