Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgpropd Structured version   Visualization version   GIF version

Theorem mulgpropd 17407
 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 3562 . . . . . . . . . . 11 (𝐵𝐾 → (𝑥𝐵𝑥𝐾))
5 ssel 3562 . . . . . . . . . . 11 (𝐵𝐾 → (𝑦𝐵𝑦𝐾))
64, 5anim12d 584 . . . . . . . . . 10 (𝐵𝐾 → ((𝑥𝐵𝑦𝐵) → (𝑥𝐾𝑦𝐾)))
73, 6syl 17 . . . . . . . . 9 (𝜑 → ((𝑥𝐵𝑦𝐵) → (𝑥𝐾𝑦𝐾)))
87imp 444 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐾𝑦𝐾))
9 mulgpropd.e . . . . . . . 8 ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
108, 9syldan 486 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
111, 2, 10grpidpropd 17084 . . . . . 6 (𝜑 → (0g𝐺) = (0g𝐻))
12113ad2ant1 1075 . . . . 5 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (0g𝐺) = (0g𝐻))
13 1zzd 11285 . . . . . . . 8 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 1 ∈ ℤ)
14 vex 3176 . . . . . . . . . . . 12 𝑏 ∈ V
1514fvconst2 6374 . . . . . . . . . . 11 (𝑥 ∈ ℕ → ((ℕ × {𝑏})‘𝑥) = 𝑏)
16 nnuz 11599 . . . . . . . . . . . 12 ℕ = (ℤ‘1)
1716eqcomi 2619 . . . . . . . . . . 11 (ℤ‘1) = ℕ
1815, 17eleq2s 2706 . . . . . . . . . 10 (𝑥 ∈ (ℤ‘1) → ((ℕ × {𝑏})‘𝑥) = 𝑏)
1918adantl 481 . . . . . . . . 9 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ 𝑥 ∈ (ℤ‘1)) → ((ℕ × {𝑏})‘𝑥) = 𝑏)
2033ad2ant1 1075 . . . . . . . . . . 11 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 𝐵𝐾)
21 simp3 1056 . . . . . . . . . . 11 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 𝑏𝐵)
2220, 21sseldd 3569 . . . . . . . . . 10 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 𝑏𝐾)
2322adantr 480 . . . . . . . . 9 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ 𝑥 ∈ (ℤ‘1)) → 𝑏𝐾)
2419, 23eqeltrd 2688 . . . . . . . 8 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ 𝑥 ∈ (ℤ‘1)) → ((ℕ × {𝑏})‘𝑥) ∈ 𝐾)
25 mulgpropd.k . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)
26253ad2antl1 1216 . . . . . . . 8 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)
2793ad2antl1 1216 . . . . . . . 8 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
2813, 24, 26, 27seqfeq3 12713 . . . . . . 7 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → seq1((+g𝐺), (ℕ × {𝑏})) = seq1((+g𝐻), (ℕ × {𝑏})))
2928fveq1d 6105 . . . . . 6 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎) = (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎))
301, 2, 10grpinvpropd 17313 . . . . . . . 8 (𝜑 → (invg𝐺) = (invg𝐻))
31303ad2ant1 1075 . . . . . . 7 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (invg𝐺) = (invg𝐻))
3228fveq1d 6105 . . . . . . 7 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎) = (seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))
3331, 32fveq12d 6109 . . . . . 6 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎)))
3429, 33ifeq12d 4056 . . . . 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 4056 . . . 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)))))
3635mpt2eq3dva 6617 . . 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 2611 . . . 4 (𝜑 → ℤ = ℤ)
38 eqidd 2611 . . . 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, 38mpt2eq123dv 6615 . . 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 2611 . . . 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, 40mpt2eq123dv 6615 . . 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 2652 . 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 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
44 eqid 2610 . . 3 (+g𝐺) = (+g𝐺)
45 eqid 2610 . . 3 (0g𝐺) = (0g𝐺)
46 eqid 2610 . . 3 (invg𝐺) = (invg𝐺)
47 mulgpropd.m . . 3 · = (.g𝐺)
4843, 44, 45, 46, 47mulgfval 17365 . 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 2610 . . 3 (Base‘𝐻) = (Base‘𝐻)
50 eqid 2610 . . 3 (+g𝐻) = (+g𝐻)
51 eqid 2610 . . 3 (0g𝐻) = (0g𝐻)
52 eqid 2610 . . 3 (invg𝐻) = (invg𝐻)
53 mulgpropd.n . . 3 × = (.g𝐻)
5449, 50, 51, 52, 53mulgfval 17365 . 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 2669 1 (𝜑· = × )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ifcif 4036  {csn 4125   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   < clt 9953  -cneg 10146  ℕcn 10897  ℤcz 11254  ℤ≥cuz 11563  seqcseq 12663  Basecbs 15695  +gcplusg 15768  0gc0g 15923  invgcminusg 17246  .gcmg 17363 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-0g 15925  df-minusg 17249  df-mulg 17364 This theorem is referenced by:  mulgass3  18460  coe1tm  19464  ply1coe  19487  evl1expd  19530
 Copyright terms: Public domain W3C validator