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Theorem isnsgrp 12647
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 995 . . . . . . 7 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑋𝐵)
2 oveq1 5860 . . . . . . . . . . . . 13 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
32oveq1d 5868 . . . . . . . . . . . 12 (𝑥 = 𝑋 → ((𝑥 𝑦) 𝑧) = ((𝑋 𝑦) 𝑧))
4 oveq1 5860 . . . . . . . . . . . 12 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
53, 4eqeq12d 2185 . . . . . . . . . . 11 (𝑥 = 𝑋 → (((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
65notbid 662 . . . . . . . . . 10 (𝑥 = 𝑋 → (¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
76rexbidv 2471 . . . . . . . . 9 (𝑥 = 𝑋 → (∃𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
87rexbidv 2471 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ∃𝑦𝐵𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
98adantl 275 . . . . . . 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 996 . . . . . . . 8 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑌𝐵)
11 oveq2 5861 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
1211oveq1d 5868 . . . . . . . . . . . 12 (𝑦 = 𝑌 → ((𝑋 𝑦) 𝑧) = ((𝑋 𝑌) 𝑧))
13 oveq1 5860 . . . . . . . . . . . . 13 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1413oveq2d 5869 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
1512, 14eqeq12d 2185 . . . . . . . . . . 11 (𝑦 = 𝑌 → (((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1615notbid 662 . . . . . . . . . 10 (𝑦 = 𝑌 → (¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1716adantl 275 . . . . . . . . 9 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑦 = 𝑌) → (¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
1817rexbidv 2471 . . . . . . . 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 997 . . . . . . . . 9 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑍𝐵)
20 oveq2 5861 . . . . . . . . . . . 12 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧) = ((𝑋 𝑌) 𝑍))
21 oveq2 5861 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
2221oveq2d 5869 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
2320, 22eqeq12d 2185 . . . . . . . . . . 11 (𝑧 = 𝑍 → (((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2423notbid 662 . . . . . . . . . 10 (𝑧 = 𝑍 → (¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2524adantl 275 . . . . . . . . 9 ((((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) ∧ 𝑧 = 𝑍) → (¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
26 neneq 2362 . . . . . . . . . 10 (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
2726adantl 275 . . . . . . . . 9 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
2819, 25, 27rspcedvd 2840 . . . . . . . 8 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑧𝐵 ¬ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)))
2910, 18, 28rspcedvd 2840 . . . . . . 7 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑦𝐵𝑧𝐵 ¬ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)))
301, 9, 29rspcedvd 2840 . . . . . 6 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
31 rexnalim 2459 . . . . . . . . 9 (∃𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3231reximi 2567 . . . . . . . 8 (∃𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ∃𝑦𝐵 ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
33 rexnalim 2459 . . . . . . . 8 (∃𝑦𝐵 ¬ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ¬ ∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3432, 33syl 14 . . . . . . 7 (∃𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ¬ ∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3534reximi 2567 . . . . . 6 (∃𝑥𝐵𝑦𝐵𝑧𝐵 ¬ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ∃𝑥𝐵 ¬ ∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
36 rexnalim 2459 . . . . . 6 (∃𝑥𝐵 ¬ ∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3730, 35, 363syl 17 . . . . 5 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
3837intnand 926 . . . 4 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
39 issgrpn0.b . . . . 5 𝐵 = (Base‘𝑀)
40 issgrpn0.o . . . . 5 = (+g𝑀)
4139, 40issgrp 12644 . . . 4 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
4238, 41sylnibr 672 . . 3 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → ¬ 𝑀 ∈ Smgrp)
43 df-nel 2436 . . 3 (𝑀 ∉ Smgrp ↔ ¬ 𝑀 ∈ Smgrp)
4442, 43sylibr 133 . 2 (((𝑋𝐵𝑌𝐵𝑍𝐵) ∧ ((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍))) → 𝑀 ∉ Smgrp)
4544ex 114 1 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (((𝑋 𝑌) 𝑍) ≠ (𝑋 (𝑌 𝑍)) → 𝑀 ∉ Smgrp))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  wcel 2141  wne 2340  wnel 2435  wral 2448  wrex 2449  cfv 5198  (class class class)co 5853  Basecbs 12416  +gcplusg 12480  Mgmcmgm 12608  Smgrpcsgrp 12642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-ov 5856  df-inn 8879  df-2 8937  df-ndx 12419  df-slot 12420  df-base 12422  df-plusg 12493  df-sgrp 12643
This theorem is referenced by: (None)
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