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Theorem issgrp 12701
Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
Hypotheses
Ref Expression
issgrp.b 𝐵 = (Base‘𝑀)
issgrp.o = (+g𝑀)
Assertion
Ref Expression
issgrp (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥, ,𝑦,𝑧

Proof of Theorem issgrp
Dummy variables 𝑏 𝑔 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 basfn 12502 . . . . 5 Base Fn V
2 vex 2740 . . . . 5 𝑔 ∈ V
3 funfvex 5528 . . . . . 6 ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V)
43funfni 5312 . . . . 5 ((Base Fn V ∧ 𝑔 ∈ V) → (Base‘𝑔) ∈ V)
51, 2, 4mp2an 426 . . . 4 (Base‘𝑔) ∈ V
65a1i 9 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) ∈ V)
7 fveq2 5511 . . . 4 (𝑔 = 𝑀 → (Base‘𝑔) = (Base‘𝑀))
8 issgrp.b . . . 4 𝐵 = (Base‘𝑀)
97, 8eqtr4di 2228 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) = 𝐵)
10 plusgslid 12551 . . . . . . 7 (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
1110slotex 12472 . . . . . 6 (𝑔 ∈ V → (+g𝑔) ∈ V)
1211elv 2741 . . . . 5 (+g𝑔) ∈ V
1312a1i 9 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) ∈ V)
14 fveq2 5511 . . . . . 6 (𝑔 = 𝑀 → (+g𝑔) = (+g𝑀))
1514adantr 276 . . . . 5 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = (+g𝑀))
16 issgrp.o . . . . 5 = (+g𝑀)
1715, 16eqtr4di 2228 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = )
18 simplr 528 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
19 id 19 . . . . . . . . . 10 (𝑜 = 𝑜 = )
20 oveq 5875 . . . . . . . . . 10 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
21 eqidd 2178 . . . . . . . . . 10 (𝑜 = 𝑧 = 𝑧)
2219, 20, 21oveq123d 5890 . . . . . . . . 9 (𝑜 = → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 𝑦) 𝑧))
23 eqidd 2178 . . . . . . . . . 10 (𝑜 = 𝑥 = 𝑥)
24 oveq 5875 . . . . . . . . . 10 (𝑜 = → (𝑦𝑜𝑧) = (𝑦 𝑧))
2519, 23, 24oveq123d 5890 . . . . . . . . 9 (𝑜 = → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 (𝑦 𝑧)))
2622, 25eqeq12d 2192 . . . . . . . 8 (𝑜 = → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2726adantl 277 . . . . . . 7 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2818, 27raleqbidv 2684 . . . . . 6 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2918, 28raleqbidv 2684 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
3018, 29raleqbidv 2684 . . . 4 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
3113, 17, 30sbcied2 3000 . . 3 ((𝑔 = 𝑀𝑏 = 𝐵) → ([(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
326, 9, 31sbcied2 3000 . 2 (𝑔 = 𝑀 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
33 df-sgrp 12700 . 2 Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
3432, 33elrab2 2896 1 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  Vcvv 2737  [wsbc 2962   Fn wfn 5207  cfv 5212  (class class class)co 5869  Basecbs 12445  +gcplusg 12518  Mgmcmgm 12665  Smgrpcsgrp 12699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-sgrp 12700
This theorem is referenced by:  issgrpv  12702  issgrpn0  12703  isnsgrp  12704  sgrpmgm  12705  sgrpass  12706  sgrp0  12707  sgrp1  12708
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