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Theorem grpodivfval 28303
Description: Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdiv.1 𝑋 = ran 𝐺
grpdiv.2 𝑁 = (inv‘𝐺)
grpdiv.3 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
grpodivfval (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦)

Proof of Theorem grpodivfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpdiv.3 . 2 𝐷 = ( /𝑔𝐺)
2 grpdiv.1 . . . . 5 𝑋 = ran 𝐺
3 rnexg 7606 . . . . 5 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
42, 3eqeltrid 2915 . . . 4 (𝐺 ∈ GrpOp → 𝑋 ∈ V)
5 mpoexga 7767 . . . 4 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V)
64, 4, 5syl2anc 586 . . 3 (𝐺 ∈ GrpOp → (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V)
7 rneq 5799 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
87, 2syl6eqr 2872 . . . . 5 (𝑔 = 𝐺 → ran 𝑔 = 𝑋)
9 id 22 . . . . . 6 (𝑔 = 𝐺𝑔 = 𝐺)
10 eqidd 2820 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
11 fveq2 6663 . . . . . . . 8 (𝑔 = 𝐺 → (inv‘𝑔) = (inv‘𝐺))
12 grpdiv.2 . . . . . . . 8 𝑁 = (inv‘𝐺)
1311, 12syl6eqr 2872 . . . . . . 7 (𝑔 = 𝐺 → (inv‘𝑔) = 𝑁)
1413fveq1d 6665 . . . . . 6 (𝑔 = 𝐺 → ((inv‘𝑔)‘𝑦) = (𝑁𝑦))
159, 10, 14oveq123d 7169 . . . . 5 (𝑔 = 𝐺 → (𝑥𝑔((inv‘𝑔)‘𝑦)) = (𝑥𝐺(𝑁𝑦)))
168, 8, 15mpoeq123dv 7221 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
17 df-gdiv 28265 . . . 4 /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
1816, 17fvmptg 6759 . . 3 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))) ∈ V) → ( /𝑔𝐺) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
196, 18mpdan 685 . 2 (𝐺 ∈ GrpOp → ( /𝑔𝐺) = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
201, 19syl5eq 2866 1 (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1531  wcel 2108  Vcvv 3493  ran crn 5549  cfv 6348  (class class class)co 7148  cmpo 7150  GrpOpcgr 28258  invcgn 28260   /𝑔 cgs 28261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-gdiv 28265
This theorem is referenced by:  grpodivval  28304  grpodivf  28307  nvmfval  28413
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