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Theorem ismgm 12781
Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
Hypotheses
Ref Expression
ismgm.b 𝐡 = (Baseβ€˜π‘€)
ismgm.o ⚬ = (+gβ€˜π‘€)
Assertion
Ref Expression
ismgm (𝑀 ∈ 𝑉 β†’ (𝑀 ∈ Mgm ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ⚬ 𝑦) ∈ 𝐡))
Distinct variable groups:   π‘₯,𝐡,𝑦   π‘₯,𝑀,𝑦   π‘₯, ⚬ ,𝑦
Allowed substitution hints:   𝑉(π‘₯,𝑦)

Proof of Theorem ismgm
Dummy variables 𝑏 π‘š π‘œ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 basfn 12522 . . . . 5 Base Fn V
2 vex 2742 . . . . 5 π‘š ∈ V
3 funfvex 5534 . . . . . 6 ((Fun Base ∧ π‘š ∈ dom Base) β†’ (Baseβ€˜π‘š) ∈ V)
43funfni 5318 . . . . 5 ((Base Fn V ∧ π‘š ∈ V) β†’ (Baseβ€˜π‘š) ∈ V)
51, 2, 4mp2an 426 . . . 4 (Baseβ€˜π‘š) ∈ V
65a1i 9 . . 3 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) ∈ V)
7 fveq2 5517 . . . 4 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = (Baseβ€˜π‘€))
8 ismgm.b . . . 4 𝐡 = (Baseβ€˜π‘€)
97, 8eqtr4di 2228 . . 3 (π‘š = 𝑀 β†’ (Baseβ€˜π‘š) = 𝐡)
10 plusgslid 12573 . . . . . . 7 (+g = Slot (+gβ€˜ndx) ∧ (+gβ€˜ndx) ∈ β„•)
1110slotex 12491 . . . . . 6 (π‘š ∈ V β†’ (+gβ€˜π‘š) ∈ V)
1211elv 2743 . . . . 5 (+gβ€˜π‘š) ∈ V
1312a1i 9 . . . 4 ((π‘š = 𝑀 ∧ 𝑏 = 𝐡) β†’ (+gβ€˜π‘š) ∈ V)
14 fveq2 5517 . . . . . 6 (π‘š = 𝑀 β†’ (+gβ€˜π‘š) = (+gβ€˜π‘€))
1514adantr 276 . . . . 5 ((π‘š = 𝑀 ∧ 𝑏 = 𝐡) β†’ (+gβ€˜π‘š) = (+gβ€˜π‘€))
16 ismgm.o . . . . 5 ⚬ = (+gβ€˜π‘€)
1715, 16eqtr4di 2228 . . . 4 ((π‘š = 𝑀 ∧ 𝑏 = 𝐡) β†’ (+gβ€˜π‘š) = ⚬ )
18 simplr 528 . . . . 5 (((π‘š = 𝑀 ∧ 𝑏 = 𝐡) ∧ π‘œ = ⚬ ) β†’ 𝑏 = 𝐡)
19 oveq 5883 . . . . . . . 8 (π‘œ = ⚬ β†’ (π‘₯π‘œπ‘¦) = (π‘₯ ⚬ 𝑦))
2019adantl 277 . . . . . . 7 (((π‘š = 𝑀 ∧ 𝑏 = 𝐡) ∧ π‘œ = ⚬ ) β†’ (π‘₯π‘œπ‘¦) = (π‘₯ ⚬ 𝑦))
2120, 18eleq12d 2248 . . . . . 6 (((π‘š = 𝑀 ∧ 𝑏 = 𝐡) ∧ π‘œ = ⚬ ) β†’ ((π‘₯π‘œπ‘¦) ∈ 𝑏 ↔ (π‘₯ ⚬ 𝑦) ∈ 𝐡))
2218, 21raleqbidv 2685 . . . . 5 (((π‘š = 𝑀 ∧ 𝑏 = 𝐡) ∧ π‘œ = ⚬ ) β†’ (βˆ€π‘¦ ∈ 𝑏 (π‘₯π‘œπ‘¦) ∈ 𝑏 ↔ βˆ€π‘¦ ∈ 𝐡 (π‘₯ ⚬ 𝑦) ∈ 𝐡))
2318, 22raleqbidv 2685 . . . 4 (((π‘š = 𝑀 ∧ 𝑏 = 𝐡) ∧ π‘œ = ⚬ ) β†’ (βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯π‘œπ‘¦) ∈ 𝑏 ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ⚬ 𝑦) ∈ 𝐡))
2413, 17, 23sbcied2 3002 . . 3 ((π‘š = 𝑀 ∧ 𝑏 = 𝐡) β†’ ([(+gβ€˜π‘š) / π‘œ]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯π‘œπ‘¦) ∈ 𝑏 ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ⚬ 𝑦) ∈ 𝐡))
256, 9, 24sbcied2 3002 . 2 (π‘š = 𝑀 β†’ ([(Baseβ€˜π‘š) / 𝑏][(+gβ€˜π‘š) / π‘œ]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯π‘œπ‘¦) ∈ 𝑏 ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ⚬ 𝑦) ∈ 𝐡))
26 df-mgm 12780 . 2 Mgm = {π‘š ∣ [(Baseβ€˜π‘š) / 𝑏][(+gβ€˜π‘š) / π‘œ]βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯π‘œπ‘¦) ∈ 𝑏}
2725, 26elab2g 2886 1 (𝑀 ∈ 𝑉 β†’ (𝑀 ∈ Mgm ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ ⚬ 𝑦) ∈ 𝐡))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  Vcvv 2739  [wsbc 2964   Fn wfn 5213  β€˜cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  Mgmcmgm 12778
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
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-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-mgm 12780
This theorem is referenced by:  ismgmn0  12782  mgmcl  12783  mgm0  12793  issgrpv  12815
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