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Theorem mulgpropdg 12978
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
mulgpropdg.m (𝜑· = (.g𝐺))
mulgpropdg.n (𝜑× = (.g𝐻))
mulgpropdg.g (𝜑𝐺𝑉)
mulgpropdg.h (𝜑𝐻𝑊)
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
mulgpropdg (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)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem mulgpropdg
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 mulgpropdg.g . . . . . . 7 (𝜑𝐺𝑉)
4 mulgpropdg.h . . . . . . 7 (𝜑𝐻𝑊)
5 mulgpropd.i . . . . . . . . . 10 (𝜑𝐵𝐾)
6 ssel 3149 . . . . . . . . . . 11 (𝐵𝐾 → (𝑥𝐵𝑥𝐾))
7 ssel 3149 . . . . . . . . . . 11 (𝐵𝐾 → (𝑦𝐵𝑦𝐾))
86, 7anim12d 335 . . . . . . . . . 10 (𝐵𝐾 → ((𝑥𝐵𝑦𝐵) → (𝑥𝐾𝑦𝐾)))
95, 8syl 14 . . . . . . . . 9 (𝜑 → ((𝑥𝐵𝑦𝐵) → (𝑥𝐾𝑦𝐾)))
109imp 124 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝐾𝑦𝐾))
11 mulgpropd.e . . . . . . . 8 ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
1210, 11syldan 282 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
131, 2, 3, 4, 12grpidpropdg 12747 . . . . . 6 (𝜑 → (0g𝐺) = (0g𝐻))
14133ad2ant1 1018 . . . . 5 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (0g𝐺) = (0g𝐻))
15 1zzd 9278 . . . . . . . 8 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 1 ∈ ℤ)
16 nnuz 9561 . . . . . . . . 9 ℕ = (ℤ‘1)
1753ad2ant1 1018 . . . . . . . . . 10 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 𝐵𝐾)
18 simp3 999 . . . . . . . . . 10 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 𝑏𝐵)
1917, 18sseldd 3156 . . . . . . . . 9 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → 𝑏𝐾)
2016, 19ialgrlemconst 12037 . . . . . . . 8 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ 𝑥 ∈ (ℤ‘1)) → ((ℕ × {𝑏})‘𝑥) ∈ 𝐾)
21 mulgpropd.k . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)
22213ad2antl1 1159 . . . . . . . 8 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) ∈ 𝐾)
23113ad2antl1 1159 . . . . . . . 8 (((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) ∧ (𝑥𝐾𝑦𝐾)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
2415, 20, 22, 23seqfeq3 10509 . . . . . . 7 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → seq1((+g𝐺), (ℕ × {𝑏})) = seq1((+g𝐻), (ℕ × {𝑏})))
2524fveq1d 5517 . . . . . 6 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎) = (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎))
261, 2, 3, 4, 12grpinvpropdg 12899 . . . . . . . 8 (𝜑 → (invg𝐺) = (invg𝐻))
27263ad2ant1 1018 . . . . . . 7 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (invg𝐺) = (invg𝐻))
2824fveq1d 5517 . . . . . . 7 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → (seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎) = (seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))
2927, 28fveq12d 5522 . . . . . 6 ((𝜑𝑎 ∈ ℤ ∧ 𝑏𝐵) → ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎)) = ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎)))
3025, 29ifeq12d 3553 . . . . 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))))
3114, 30ifeq12d 3553 . . . 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)))))
3231mpoeq3dva 5938 . . 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))))))
33 eqidd 2178 . . . 4 (𝜑 → ℤ = ℤ)
34 eqidd 2178 . . . 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)))))
3533, 1, 34mpoeq123dv 5936 . . 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))))))
36 eqidd 2178 . . . 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)))))
3733, 2, 36mpoeq123dv 5936 . . 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))))))
3832, 35, 373eqtr3d 2218 . 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))))))
39 mulgpropdg.m . . 3 (𝜑· = (.g𝐺))
40 eqid 2177 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
41 eqid 2177 . . . . 5 (+g𝐺) = (+g𝐺)
42 eqid 2177 . . . . 5 (0g𝐺) = (0g𝐺)
43 eqid 2177 . . . . 5 (invg𝐺) = (invg𝐺)
44 eqid 2177 . . . . 5 (.g𝐺) = (.g𝐺)
4540, 41, 42, 43, 44mulgfvalg 12939 . . . 4 (𝐺𝑉 → (.g𝐺) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎))))))
463, 45syl 14 . . 3 (𝜑 → (.g𝐺) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g𝐺), if(0 < 𝑎, (seq1((+g𝐺), (ℕ × {𝑏}))‘𝑎), ((invg𝐺)‘(seq1((+g𝐺), (ℕ × {𝑏}))‘-𝑎))))))
4739, 46eqtrd 2210 . 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))))))
48 mulgpropdg.n . . 3 (𝜑× = (.g𝐻))
49 eqid 2177 . . . . 5 (Base‘𝐻) = (Base‘𝐻)
50 eqid 2177 . . . . 5 (+g𝐻) = (+g𝐻)
51 eqid 2177 . . . . 5 (0g𝐻) = (0g𝐻)
52 eqid 2177 . . . . 5 (invg𝐻) = (invg𝐻)
53 eqid 2177 . . . . 5 (.g𝐻) = (.g𝐻)
5449, 50, 51, 52, 53mulgfvalg 12939 . . . 4 (𝐻𝑊 → (.g𝐻) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))))))
554, 54syl 14 . . 3 (𝜑 → (.g𝐻) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))))))
5648, 55eqtrd 2210 . 2 (𝜑× = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g𝐻), if(0 < 𝑎, (seq1((+g𝐻), (ℕ × {𝑏}))‘𝑎), ((invg𝐻)‘(seq1((+g𝐻), (ℕ × {𝑏}))‘-𝑎))))))
5738, 47, 563eqtr4d 2220 1 (𝜑· = × )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wss 3129  ifcif 3534  {csn 3592   class class class wbr 4003   × cxp 4624  cfv 5216  (class class class)co 5874  cmpo 5876  0cc0 7810  1c1 7811   < clt 7990  -cneg 8127  cn 8917  cz 9251  seqcseq 10442  Basecbs 12456  +gcplusg 12530  0gc0g 12695  invgcminusg 12832  .gcmg 12937
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-in1 614  ax-in2 615  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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-seqfrec 10443  df-ndx 12459  df-slot 12460  df-base 12462  df-0g 12697  df-minusg 12835  df-mulg 12938
This theorem is referenced by:  mulgass3  13207
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