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Theorem grpsubfvalALT 19002
Description: Shorter proof of grpsubfval 19001 using ax-rep 5279. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpsubval.b 𝐵 = (Base‘𝐺)
grpsubval.p + = (+g𝐺)
grpsubval.i 𝐼 = (invg𝐺)
grpsubval.m = (-g𝐺)
Assertion
Ref Expression
grpsubfvalALT = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝐼,𝑦   𝑥, + ,𝑦
Allowed substitution hints:   (𝑥,𝑦)

Proof of Theorem grpsubfvalALT
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 grpsubval.m . . 3 = (-g𝐺)
2 fveq2 6906 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpsubval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2795 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6906 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpsubval.p . . . . . . 7 + = (+g𝐺)
75, 6eqtr4di 2795 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
8 eqidd 2738 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
9 fveq2 6906 . . . . . . . 8 (𝑔 = 𝐺 → (invg𝑔) = (invg𝐺))
10 grpsubval.i . . . . . . . 8 𝐼 = (invg𝐺)
119, 10eqtr4di 2795 . . . . . . 7 (𝑔 = 𝐺 → (invg𝑔) = 𝐼)
1211fveq1d 6908 . . . . . 6 (𝑔 = 𝐺 → ((invg𝑔)‘𝑦) = (𝐼𝑦))
137, 8, 12oveq123d 7452 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)((invg𝑔)‘𝑦)) = (𝑥 + (𝐼𝑦)))
144, 4, 13mpoeq123dv 7508 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
15 df-sbg 18956 . . . 4 -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
163fvexi 6920 . . . . 5 𝐵 ∈ V
1716, 16mpoex 8104 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) ∈ V
1814, 15, 17fvmpt 7016 . . 3 (𝐺 ∈ V → (-g𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
191, 18eqtrid 2789 . 2 (𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
20 fvprc 6898 . . . 4 𝐺 ∈ V → (-g𝐺) = ∅)
211, 20eqtrid 2789 . . 3 𝐺 ∈ V → = ∅)
22 fvprc 6898 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
233, 22eqtrid 2789 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
2423olcd 875 . . . 4 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
25 0mpo0 7516 . . . 4 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
2624, 25syl 17 . . 3 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
2721, 26eqtr4d 2780 . 2 𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
2819, 27pm2.61i 182 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 848   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  +gcplusg 17297  invgcminusg 18952  -gcsg 18953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-sbg 18956
This theorem is referenced by: (None)
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