MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnsgrp Structured version   Visualization version   GIF version

Theorem isnsgrp 17481
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 ∉ SGrp))

Proof of Theorem isnsgrp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1225 . . . . . . 7 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑋𝐵)
2 oveq1 6812 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
32oveq1d 6820 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ((𝑥 𝑦) 𝑧) = ((𝑋 𝑦) 𝑧))
4 oveq1 6812 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
53, 4eqeq12d 2767 . . . . . . . . . . 11 (𝑥 = 𝑋 → (((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
65notbid 307 . . . . . . . . . 10 (𝑥 = 𝑋 → (¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
76rexbidv 3182 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
87rexbidv 3182 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑦𝐵𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
98adantl 473 . . . . . . 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 1227 . . . . . . . 8 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑌𝐵)
11 oveq2 6813 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
1211oveq1d 6820 . . . . . . . . . . . 12 (𝑦 = 𝑌 → ((𝑋 𝑦) 𝑧) = ((𝑋 𝑌) 𝑧))
13 oveq1 6812 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1413oveq2d 6821 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
1512, 14eqeq12d 2767 . . . . . . . . . . 11 (𝑦 = 𝑌 → (((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1615notbid 307 . . . . . . . . . 10 (𝑦 = 𝑌 → (¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1716adantl 473 . . . . . . . . 9 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑦 = 𝑌) → (¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1817rexbidv 3182 . . . . . . . 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 1229 . . . . . . . . 9 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑍𝐵)
20 oveq2 6813 . . . . . . . . . . . 12 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧) = ((𝑋 𝑌) 𝑍))
21 oveq2 6813 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
2221oveq2d 6821 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
2320, 22eqeq12d 2767 . . . . . . . . . . 11 (𝑧 = 𝑍 → (((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2423notbid 307 . . . . . . . . . 10 (𝑧 = 𝑍 → (¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2524adantl 473 . . . . . . . . 9 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑧 = 𝑍) → (¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
26 neneq 2930 . . . . . . . . . 10 (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
2726adantl 473 . . . . . . . . 9 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
2819, 25, 27rspcedvd 3448 . . . . . . . 8 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑧𝐵 ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)))
2910, 18, 28rspcedvd 3448 . . . . . . 7 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑦𝐵𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)))
301, 9, 29rspcedvd 3448 . . . . . 6 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
31 rexnal 3125 . . . . . . . 8 (∃𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
32312rexbii 3172 . . . . . . 7 (∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑥𝐵𝑦𝐵 ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
33 rexnal2 3173 . . . . . . 7 (∃𝑥𝐵𝑦𝐵 ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3432, 33bitr2i 265 . . . . . 6 (¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3530, 34sylibr 224 . . . . 5 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3635intnand 1000 . . . 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 17478 . . . 4 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
4036, 39sylnibr 318 . . 3 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ 𝑀 ∈ SGrp)
41 df-nel 3028 . . 3 (𝑀 ∉ SGrp ↔ ¬ 𝑀 ∈ SGrp)
4240, 41sylibr 224 . 2 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑀 ∉ SGrp)
4342ex 449 1 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → 𝑀 ∉ SGrp))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wnel 3027  wral 3042  wrex 3043  cfv 6041  (class class class)co 6805  Basecbs 16051  +gcplusg 16135  Mgmcmgm 17433  SGrpcsgrp 17476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-nul 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-iota 6004  df-fv 6049  df-ov 6808  df-sgrp 17477
This theorem is referenced by:  mgm2nsgrplem4  17601  xrsnsgrp  19976
  Copyright terms: Public domain W3C validator