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Theorem mulgpropd 18201
Description: Two structures with the same group-nature have the same group multiple function. 𝐾 is expected to either be V (when strong equality is available) or 𝐵 (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
mulgpropd.m · = (.g𝐺)
mulgpropd.n × = (.g𝐻)
mulgpropd.b1 (𝜑𝐵 = (Base‘𝐺))
mulgpropd.b2 (𝜑𝐵 = (Base‘𝐻))
mulgpropd.i (𝜑𝐵𝐾)
mulgpropd.k ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)
mulgpropd.e ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
Assertion
Ref Expression
mulgpropd (𝜑· = × )
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝐾,𝑦
Allowed substitution hints:   · (𝑥,𝑦)   × (𝑥,𝑦)

Proof of Theorem mulgpropd
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgpropd.b1 . . . . . . 7 (𝜑𝐵 = (Base‘𝐺))
2 mulgpropd.b2 . . . . . . 7 (𝜑𝐵 = (Base‘𝐻))
3 mulgpropd.i . . . . . . . . . 10 (𝜑𝐵𝐾)
4 ssel 3964 . . . . . . . . . . 11 (𝐵𝐾 → (𝑥𝐵𝑥𝐾))
5 ssel 3964 . . . . . . . . . . 11 (𝐵𝐾 → (𝑦𝐵𝑦𝐾))
64, 5anim12d 608 . . . . . . . . . 10 (𝐵𝐾 → ((𝑥𝐵𝑦𝐵) → (𝑥𝐾𝑦𝐾)))
73, 6syl 17 . . . . . . . . 9 (𝜑 → ((𝑥𝐵𝑦𝐵) → (𝑥𝐾𝑦𝐾)))
87imp 407 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐾𝑦𝐾))
9 mulgpropd.e . . . . . . . 8 ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
108, 9syldan 591 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
111, 2, 10grpidpropd 17863 . . . . . 6 (𝜑 → (0g𝐺) = (0g𝐻))
12113ad2ant1 1127 . . . . 5 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (0g𝐺) = (0g𝐻))
13 1zzd 12005 . . . . . . . 8 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 1 ∈ ℤ)
14 vex 3502 . . . . . . . . . . . 12 𝑏 ∈ V
1514fvconst2 6965 . . . . . . . . . . 11 (𝑥 ∈ ℕ → ((ℕ × {𝑏})‘𝑥) = 𝑏)
16 nnuz 12273 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
1716eqcomi 2834 . . . . . . . . . . 11 (ℤ‘1) = ℕ
1815, 17eleq2s 2935 . . . . . . . . . 10 (𝑥 ∈ (ℤ‘1) → ((ℕ × {𝑏})‘𝑥) = 𝑏)
1918adantl 482 . . . . . . . . 9 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ 𝑥 ∈ (ℤ‘1)) → ((ℕ × {𝑏})‘𝑥) = 𝑏)
2033ad2ant1 1127 . . . . . . . . . . 11 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 𝐵𝐾)
21 simp3 1132 . . . . . . . . . . 11 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 𝑏𝐵)
2220, 21sseldd 3971 . . . . . . . . . 10 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 𝑏𝐾)
2322adantr 481 . . . . . . . . 9 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ 𝑥 ∈ (ℤ‘1)) → 𝑏𝐾)
2419, 23eqeltrd 2917 . . . . . . . 8 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ 𝑥 ∈ (ℤ‘1)) → ((ℕ × {𝑏})‘𝑥) ∈ 𝐾)
25 mulgpropd.k . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)
26253ad2antl1 1179 . . . . . . . 8 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)
2793ad2antl1 1179 . . . . . . . 8 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
2813, 24, 26, 27seqfeq3 13413 . . . . . . 7 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → seq1((+g𝐺), (ℕ × {𝑏})) = seq1((+g𝐻), (ℕ × {𝑏})))
2928fveq1d 6668 . . . . . 6 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎) = (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎))
301, 2, 10grpinvpropd 18106 . . . . . . . 8 (𝜑 → (invg𝐺) = (invg𝐻))
31303ad2ant1 1127 . . . . . . 7 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (invg𝐺) = (invg𝐻))
3228fveq1d 6668 . . . . . . 7 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎) = (seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))
3331, 32fveq12d 6673 . . . . . 6 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎)))
3429, 33ifeq12d 4489 . . . . 5 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎))) = if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))))
3512, 34ifeq12d 4489 . . . 4 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎)))) = if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎)))))
3635mpoeq3dva 7226 . . 3 (𝜑 → (𝑎 ∈ ℤ, 𝑏𝐵 ↦ if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎))))) = (𝑎 ∈ ℤ, 𝑏𝐵 ↦ if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))))))
37 eqidd 2826 . . . 4 (𝜑 → ℤ = ℤ)
38 eqidd 2826 . . . 4 (𝜑 → if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎)))) = if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎)))))
3937, 1, 38mpoeq123dv 7224 . . 3 (𝜑 → (𝑎 ∈ ℤ, 𝑏𝐵 ↦ if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎))))) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎))))))
40 eqidd 2826 . . . 4 (𝜑 → if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎)))) = if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎)))))
4137, 2, 40mpoeq123dv 7224 . . 3 (𝜑 → (𝑎 ∈ ℤ, 𝑏𝐵 ↦ if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))))) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))))))
4236, 39, 413eqtr3d 2868 . 2 (𝜑 → (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎))))) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))))))
43 eqid 2825 . . 3 (Base‘𝐺) = (Base‘𝐺)
44 eqid 2825 . . 3 (+g𝐺) = (+g𝐺)
45 eqid 2825 . . 3 (0g𝐺) = (0g𝐺)
46 eqid 2825 . . 3 (invg𝐺) = (invg𝐺)
47 mulgpropd.m . . 3 · = (.g𝐺)
4843, 44, 45, 46, 47mulgfval 18158 . 2 · = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎)))))
49 eqid 2825 . . 3 (Base‘𝐻) = (Base‘𝐻)
50 eqid 2825 . . 3 (+g𝐻) = (+g𝐻)
51 eqid 2825 . . 3 (0g𝐻) = (0g𝐻)
52 eqid 2825 . . 3 (invg𝐻) = (invg𝐻)
53 mulgpropd.n . . 3 × = (.g𝐻)
5449, 50, 51, 52, 53mulgfval 18158 . 2 × = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎)))))
5542, 48, 543eqtr4g 2885 1 (𝜑· = × )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wss 3939  ifcif 4469  {csn 4563   class class class wbr 5062   × cxp 5551  cfv 6351  (class class class)co 7151  cmpo 7153  0cc0 10529  1c1 10530   < clt 10667  -cneg 10863  cn 11630  cz 11973  cuz 12235  seqcseq 13362  Basecbs 16475  +gcplusg 16557  0gc0g 16705  invgcminusg 18036  .gcmg 18156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-seq 13363  df-0g 16707  df-minusg 18039  df-mulg 18157
This theorem is referenced by:  mulgass3  19309  coe1tm  20360  ply1coe  20383  evl1expd  20426
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