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Theorem grpsubfvalg 12805
Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.)
Hypotheses
Ref Expression
grpsubval.b 𝐵 = (Base‘𝐺)
grpsubval.p + = (+g𝐺)
grpsubval.i 𝐼 = (invg𝐺)
grpsubval.m = (-g𝐺)
Assertion
Ref Expression
grpsubfvalg (𝐺𝑉 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝐼,𝑦   𝑥, + ,𝑦
Allowed substitution hints:   (𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem grpsubfvalg
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpsubval.m . 2 = (-g𝐺)
2 df-sbg 12769 . . 3 -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
3 fveq2 5511 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 grpsubval.b . . . . 5 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2228 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6 fveq2 5511 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
7 grpsubval.p . . . . . 6 + = (+g𝐺)
86, 7eqtr4di 2228 . . . . 5 (𝑔 = 𝐺 → (+g𝑔) = + )
9 eqidd 2178 . . . . 5 (𝑔 = 𝐺𝑥 = 𝑥)
10 fveq2 5511 . . . . . . 7 (𝑔 = 𝐺 → (invg𝑔) = (invg𝐺))
11 grpsubval.i . . . . . . 7 𝐼 = (invg𝐺)
1210, 11eqtr4di 2228 . . . . . 6 (𝑔 = 𝐺 → (invg𝑔) = 𝐼)
1312fveq1d 5513 . . . . 5 (𝑔 = 𝐺 → ((invg𝑔)‘𝑦) = (𝐼𝑦))
148, 9, 13oveq123d 5890 . . . 4 (𝑔 = 𝐺 → (𝑥(+g𝑔)((invg𝑔)‘𝑦)) = (𝑥 + (𝐼𝑦)))
155, 5, 14mpoeq123dv 5931 . . 3 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
16 elex 2748 . . 3 (𝐺𝑉𝐺 ∈ V)
17 basfn 12499 . . . . . 6 Base Fn V
18 funfvex 5528 . . . . . . 7 ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
1918funfni 5312 . . . . . 6 ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
2017, 16, 19sylancr 414 . . . . 5 (𝐺𝑉 → (Base‘𝐺) ∈ V)
214, 20eqeltrid 2264 . . . 4 (𝐺𝑉𝐵 ∈ V)
22 mpoexga 6207 . . . 4 ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) ∈ V)
2321, 21, 22syl2anc 411 . . 3 (𝐺𝑉 → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) ∈ V)
242, 15, 16, 23fvmptd3 5605 . 2 (𝐺𝑉 → (-g𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
251, 24eqtrid 2222 1 (𝐺𝑉 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737   Fn wfn 5207  cfv 5212  (class class class)co 5869  cmpo 5871  Basecbs 12442  +gcplusg 12515  invgcminusg 12765  -gcsg 12766
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-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7890  ax-resscn 7891  ax-1re 7893  ax-addrcl 7896
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-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  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-iun 3886  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-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-inn 8906  df-ndx 12445  df-slot 12446  df-base 12448  df-sbg 12769
This theorem is referenced by:  grpsubval  12806  grpsubf  12835  grpsubpropdg  12860  grpsubpropd2  12861
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