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Theorem grpsubfvalALT 18266
Description: Shorter proof of grpsubfval 18265 using ax-rep 5154. (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 6674 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
3 grpsubval.b . . . . . 6 𝐵 = (Base‘𝐺)
42, 3eqtr4di 2791 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
5 fveq2 6674 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
6 grpsubval.p . . . . . . 7 + = (+g𝐺)
75, 6eqtr4di 2791 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
8 eqidd 2739 . . . . . 6 (𝑔 = 𝐺𝑥 = 𝑥)
9 fveq2 6674 . . . . . . . 8 (𝑔 = 𝐺 → (invg𝑔) = (invg𝐺))
10 grpsubval.i . . . . . . . 8 𝐼 = (invg𝐺)
119, 10eqtr4di 2791 . . . . . . 7 (𝑔 = 𝐺 → (invg𝑔) = 𝐼)
1211fveq1d 6676 . . . . . 6 (𝑔 = 𝐺 → ((invg𝑔)‘𝑦) = (𝐼𝑦))
137, 8, 12oveq123d 7191 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)((invg𝑔)‘𝑦)) = (𝑥 + (𝐼𝑦)))
144, 4, 13mpoeq123dv 7243 . . . 4 (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
15 df-sbg 18224 . . . 4 -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g𝑔)((invg𝑔)‘𝑦))))
163fvexi 6688 . . . . 5 𝐵 ∈ V
1716, 16mpoex 7803 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) ∈ V
1814, 15, 17fvmpt 6775 . . 3 (𝐺 ∈ V → (-g𝐺) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
191, 18syl5eq 2785 . 2 (𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
20 fvprc 6666 . . . 4 𝐺 ∈ V → (-g𝐺) = ∅)
211, 20syl5eq 2785 . . 3 𝐺 ∈ V → = ∅)
22 fvprc 6666 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
233, 22syl5eq 2785 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
2423olcd 873 . . . 4 𝐺 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
25 0mpo0 7251 . . . 4 ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
2624, 25syl 17 . . 3 𝐺 ∈ V → (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))) = ∅)
2721, 26eqtr4d 2776 . 2 𝐺 ∈ V → = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦))))
2819, 27pm2.61i 185 1 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 846   = wceq 1542  wcel 2114  Vcvv 3398  c0 4211  cfv 6339  (class class class)co 7170  cmpo 7172  Basecbs 16586  +gcplusg 16668  invgcminusg 18220  -gcsg 18221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-sbg 18224
This theorem is referenced by: (None)
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