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Theorem isnsgrp 17727
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 1184 . . . . . . 7 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑋𝐵)
2 oveq1 7023 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
32oveq1d 7031 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ((𝑥 𝑦) 𝑧) = ((𝑋 𝑦) 𝑧))
4 oveq1 7023 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
53, 4eqeq12d 2810 . . . . . . . . . . 11 (𝑥 = 𝑋 → (((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
65notbid 319 . . . . . . . . . 10 (𝑥 = 𝑋 → (¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
76rexbidv 3260 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
87rexbidv 3260 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑦𝐵𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
98adantl 482 . . . . . . 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 1185 . . . . . . . 8 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑌𝐵)
11 oveq2 7024 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
1211oveq1d 7031 . . . . . . . . . . . 12 (𝑦 = 𝑌 → ((𝑋 𝑦) 𝑧) = ((𝑋 𝑌) 𝑧))
13 oveq1 7023 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1413oveq2d 7032 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
1512, 14eqeq12d 2810 . . . . . . . . . . 11 (𝑦 = 𝑌 → (((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1615notbid 319 . . . . . . . . . 10 (𝑦 = 𝑌 → (¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1716adantl 482 . . . . . . . . 9 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑦 = 𝑌) → (¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1817rexbidv 3260 . . . . . . . 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 1186 . . . . . . . . 9 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑍𝐵)
20 oveq2 7024 . . . . . . . . . . . 12 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧) = ((𝑋 𝑌) 𝑍))
21 oveq2 7024 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
2221oveq2d 7032 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
2320, 22eqeq12d 2810 . . . . . . . . . . 11 (𝑧 = 𝑍 → (((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2423notbid 319 . . . . . . . . . 10 (𝑧 = 𝑍 → (¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2524adantl 482 . . . . . . . . 9 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑧 = 𝑍) → (¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
26 neneq 2990 . . . . . . . . . 10 (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
2726adantl 482 . . . . . . . . 9 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
2819, 25, 27rspcedvd 3566 . . . . . . . 8 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑧𝐵 ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)))
2910, 18, 28rspcedvd 3566 . . . . . . 7 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑦𝐵𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)))
301, 9, 29rspcedvd 3566 . . . . . 6 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
31 rexnal 3202 . . . . . . . 8 (∃𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
32312rexbii 3212 . . . . . . 7 (∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑥𝐵𝑦𝐵 ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
33 rexnal2 3222 . . . . . . 7 (∃𝑥𝐵𝑦𝐵 ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3432, 33bitr2i 277 . . . . . 6 (¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3530, 34sylibr 235 . . . . 5 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3635intnand 489 . . . 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 17724 . . . 4 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
4036, 39sylnibr 330 . . 3 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ 𝑀 ∈ SGrp)
41 df-nel 3091 . . 3 (𝑀 ∉ SGrp ↔ ¬ 𝑀 ∈ SGrp)
4240, 41sylibr 235 . 2 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑀 ∉ SGrp)
4342ex 413 1 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → 𝑀 ∉ SGrp))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wnel 3090  wral 3105  wrex 3106  cfv 6225  (class class class)co 7016  Basecbs 16312  +gcplusg 16394  Mgmcmgm 17679  SGrpcsgrp 17722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-nul 5101
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-iota 6189  df-fv 6233  df-ov 7019  df-sgrp 17723
This theorem is referenced by:  mgm2nsgrplem4  17847  xrsnsgrp  20263
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