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Theorem isgrpda 36818
Description: Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
isgrpda.1 (πœ‘ β†’ 𝑋 ∈ V)
isgrpda.2 (πœ‘ β†’ 𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹)
isgrpda.3 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)))
isgrpda.4 (πœ‘ β†’ π‘ˆ ∈ 𝑋)
isgrpda.5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (π‘ˆπΊπ‘₯) = π‘₯)
isgrpda.6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘› ∈ 𝑋 (𝑛𝐺π‘₯) = π‘ˆ)
Assertion
Ref Expression
isgrpda (πœ‘ β†’ 𝐺 ∈ GrpOp)
Distinct variable groups:   πœ‘,π‘₯,𝑦,𝑧   𝑛,𝐺,π‘₯,𝑦,𝑧   𝑛,𝑋,π‘₯,𝑦,𝑧   π‘ˆ,𝑛,π‘₯,𝑦,𝑧
Allowed substitution hint:   πœ‘(𝑛)

Proof of Theorem isgrpda
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 isgrpda.2 . . 3 (πœ‘ β†’ 𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹)
2 isgrpda.3 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) β†’ ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)))
32ralrimivvva 3203 . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)))
4 isgrpda.4 . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑋)
5 isgrpda.5 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (π‘ˆπΊπ‘₯) = π‘₯)
6 isgrpda.6 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘› ∈ 𝑋 (𝑛𝐺π‘₯) = π‘ˆ)
7 oveq1 7415 . . . . . . . . 9 (𝑦 = 𝑛 β†’ (𝑦𝐺π‘₯) = (𝑛𝐺π‘₯))
87eqeq1d 2734 . . . . . . . 8 (𝑦 = 𝑛 β†’ ((𝑦𝐺π‘₯) = π‘ˆ ↔ (𝑛𝐺π‘₯) = π‘ˆ))
98cbvrexvw 3235 . . . . . . 7 (βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ ↔ βˆƒπ‘› ∈ 𝑋 (𝑛𝐺π‘₯) = π‘ˆ)
106, 9sylibr 233 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)
115, 10jca 512 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘ˆπΊπ‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))
1211ralrimiva 3146 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 ((π‘ˆπΊπ‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))
13 oveq1 7415 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (𝑒𝐺π‘₯) = (π‘ˆπΊπ‘₯))
1413eqeq1d 2734 . . . . . . 7 (𝑒 = π‘ˆ β†’ ((𝑒𝐺π‘₯) = π‘₯ ↔ (π‘ˆπΊπ‘₯) = π‘₯))
15 eqeq2 2744 . . . . . . . 8 (𝑒 = π‘ˆ β†’ ((𝑦𝐺π‘₯) = 𝑒 ↔ (𝑦𝐺π‘₯) = π‘ˆ))
1615rexbidv 3178 . . . . . . 7 (𝑒 = π‘ˆ β†’ (βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒 ↔ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ))
1714, 16anbi12d 631 . . . . . 6 (𝑒 = π‘ˆ β†’ (((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒) ↔ ((π‘ˆπΊπ‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
1817ralbidv 3177 . . . . 5 (𝑒 = π‘ˆ β†’ (βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒) ↔ βˆ€π‘₯ ∈ 𝑋 ((π‘ˆπΊπ‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)))
1918rspcev 3612 . . . 4 ((π‘ˆ ∈ 𝑋 ∧ βˆ€π‘₯ ∈ 𝑋 ((π‘ˆπΊπ‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = π‘ˆ)) β†’ βˆƒπ‘’ ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒))
204, 12, 19syl2anc 584 . . 3 (πœ‘ β†’ βˆƒπ‘’ ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒))
214adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ π‘ˆ ∈ 𝑋)
22 simpr 485 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
235eqcomd 2738 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ = (π‘ˆπΊπ‘₯))
24 rspceov 7455 . . . . . . . . . 10 ((π‘ˆ ∈ 𝑋 ∧ π‘₯ ∈ 𝑋 ∧ π‘₯ = (π‘ˆπΊπ‘₯)) β†’ βˆƒπ‘¦ ∈ 𝑋 βˆƒπ‘§ ∈ 𝑋 π‘₯ = (𝑦𝐺𝑧))
2521, 22, 23, 24syl3anc 1371 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝑋 βˆƒπ‘§ ∈ 𝑋 π‘₯ = (𝑦𝐺𝑧))
2625ralrimiva 3146 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝑋 βˆƒπ‘§ ∈ 𝑋 π‘₯ = (𝑦𝐺𝑧))
27 foov 7580 . . . . . . . 8 (𝐺:(𝑋 Γ— 𝑋)–onto→𝑋 ↔ (𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝑋 βˆƒπ‘§ ∈ 𝑋 π‘₯ = (𝑦𝐺𝑧)))
281, 26, 27sylanbrc 583 . . . . . . 7 (πœ‘ β†’ 𝐺:(𝑋 Γ— 𝑋)–onto→𝑋)
29 forn 6808 . . . . . . 7 (𝐺:(𝑋 Γ— 𝑋)–onto→𝑋 β†’ ran 𝐺 = 𝑋)
3028, 29syl 17 . . . . . 6 (πœ‘ β†’ ran 𝐺 = 𝑋)
3130sqxpeqd 5708 . . . . 5 (πœ‘ β†’ (ran 𝐺 Γ— ran 𝐺) = (𝑋 Γ— 𝑋))
3231, 30feq23d 6712 . . . 4 (πœ‘ β†’ (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ↔ 𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹))
3330raleqdv 3325 . . . . . 6 (πœ‘ β†’ (βˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ↔ βˆ€π‘§ ∈ 𝑋 ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧))))
3430, 33raleqbidv 3342 . . . . 5 (πœ‘ β†’ (βˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ↔ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧))))
3530, 34raleqbidv 3342 . . . 4 (πœ‘ β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧))))
3630rexeqdv 3326 . . . . . . 7 (πœ‘ β†’ (βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑒 ↔ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒))
3736anbi2d 629 . . . . . 6 (πœ‘ β†’ (((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑒) ↔ ((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒)))
3830, 37raleqbidv 3342 . . . . 5 (πœ‘ β†’ (βˆ€π‘₯ ∈ ran 𝐺((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑒) ↔ βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒)))
3930, 38rexeqbidv 3343 . . . 4 (πœ‘ β†’ (βˆƒπ‘’ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑒) ↔ βˆƒπ‘’ ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒)))
4032, 35, 393anbi123d 1436 . . 3 (πœ‘ β†’ ((𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘’ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑒)) ↔ (𝐺:(𝑋 Γ— 𝑋)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 ((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘’ ∈ 𝑋 βˆ€π‘₯ ∈ 𝑋 ((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ 𝑋 (𝑦𝐺π‘₯) = 𝑒))))
411, 3, 20, 40mpbir3and 1342 . 2 (πœ‘ β†’ (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘’ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑒)))
42 isgrpda.1 . . . . 5 (πœ‘ β†’ 𝑋 ∈ V)
4342, 42xpexd 7737 . . . 4 (πœ‘ β†’ (𝑋 Γ— 𝑋) ∈ V)
441, 43fexd 7228 . . 3 (πœ‘ β†’ 𝐺 ∈ V)
45 eqid 2732 . . . 4 ran 𝐺 = ran 𝐺
4645isgrpo 29745 . . 3 (𝐺 ∈ V β†’ (𝐺 ∈ GrpOp ↔ (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘’ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑒))))
4744, 46syl 17 . 2 (πœ‘ β†’ (𝐺 ∈ GrpOp ↔ (𝐺:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺((π‘₯𝐺𝑦)𝐺𝑧) = (π‘₯𝐺(𝑦𝐺𝑧)) ∧ βˆƒπ‘’ ∈ ran πΊβˆ€π‘₯ ∈ ran 𝐺((𝑒𝐺π‘₯) = π‘₯ ∧ βˆƒπ‘¦ ∈ ran 𝐺(𝑦𝐺π‘₯) = 𝑒))))
4841, 47mpbird 256 1 (πœ‘ β†’ 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   Γ— cxp 5674  ran crn 5677  βŸΆwf 6539  β€“ontoβ†’wfo 6541  (class class class)co 7408  GrpOpcgr 29737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-grpo 29741
This theorem is referenced by:  isdrngo2  36821
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