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Theorem issgrpd 18660
Description: Deduce a semigroup from its properties. (Contributed by AV, 13-Feb-2025.)
Hypotheses
Ref Expression
issgrpd.b (𝜑𝐵 = (Base‘𝐺))
issgrpd.p (𝜑+ = (+g𝐺))
issgrpd.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
issgrpd.a ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
issgrpd.v (𝜑𝐺𝑉)
Assertion
Ref Expression
issgrpd (𝜑𝐺 ∈ Smgrp)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐺,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   + (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem issgrpd
StepHypRef Expression
1 issgrpd.c . . . . . . 7 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
213expib 1119 . . . . . 6 (𝜑 → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵))
3 issgrpd.b . . . . . . . 8 (𝜑𝐵 = (Base‘𝐺))
43eleq2d 2813 . . . . . . 7 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐺)))
53eleq2d 2813 . . . . . . 7 (𝜑 → (𝑦𝐵𝑦 ∈ (Base‘𝐺)))
64, 5anbi12d 630 . . . . . 6 (𝜑 → ((𝑥𝐵𝑦𝐵) ↔ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))))
7 issgrpd.p . . . . . . . 8 (𝜑+ = (+g𝐺))
87oveqd 7421 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
98, 3eleq12d 2821 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
102, 6, 93imtr3d 293 . . . . 5 (𝜑 → ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺)))
1110imp 406 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
12 df-3an 1086 . . . . . . . . 9 ((𝑥𝐵𝑦𝐵𝑧𝐵) ↔ ((𝑥𝐵𝑦𝐵) ∧ 𝑧𝐵))
13 issgrpd.a . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
1412, 13sylan2br 594 . . . . . . . 8 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ 𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
1514ex 412 . . . . . . 7 (𝜑 → (((𝑥𝐵𝑦𝐵) ∧ 𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
163eleq2d 2813 . . . . . . . 8 (𝜑 → (𝑧𝐵𝑧 ∈ (Base‘𝐺)))
176, 16anbi12d 630 . . . . . . 7 (𝜑 → (((𝑥𝐵𝑦𝐵) ∧ 𝑧𝐵) ↔ ((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑧 ∈ (Base‘𝐺))))
18 eqidd 2727 . . . . . . . . 9 (𝜑𝑧 = 𝑧)
197, 8, 18oveq123d 7425 . . . . . . . 8 (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧))
20 eqidd 2727 . . . . . . . . 9 (𝜑𝑥 = 𝑥)
217oveqd 7421 . . . . . . . . 9 (𝜑 → (𝑦 + 𝑧) = (𝑦(+g𝐺)𝑧))
227, 20, 21oveq123d 7425 . . . . . . . 8 (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))
2319, 22eqeq12d 2742 . . . . . . 7 (𝜑 → (((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2415, 17, 233imtr3d 293 . . . . . 6 (𝜑 → (((𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧))))
2524impl 455 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))
2625ralrimiva 3140 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))
2711, 26jca 511 . . 3 ((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))))
2827ralrimivva 3194 . 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))))
29 issgrpd.v . . 3 (𝜑𝐺𝑉)
30 eqid 2726 . . . 4 (Base‘𝐺) = (Base‘𝐺)
31 eqid 2726 . . . 4 (+g𝐺) = (+g𝐺)
3230, 31issgrpv 18651 . . 3 (𝐺𝑉 → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))))
3329, 32syl 17 . 2 (𝜑 → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g𝐺)𝑦)(+g𝐺)𝑧) = (𝑥(+g𝐺)(𝑦(+g𝐺)𝑧)))))
3428, 33mpbird 257 1 (𝜑𝐺 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  cfv 6536  (class class class)co 7404  Basecbs 17150  +gcplusg 17203  Smgrpcsgrp 18648
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-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-mgm 18570  df-sgrp 18649
This theorem is referenced by:  prdssgrpd  18663  isrngd  20075
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