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 → · = (.g‘𝐺))
mulgpropdg.n	⊢ (𝜑 → × = (.g‘𝐻))
mulgpropdg.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
mulgpropdg.h	⊢ (𝜑 → 𝐻 ∈ 𝑊)
mulgpropd.b1	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
mulgpropd.b2	⊢ (𝜑 → 𝐵 = (Base‘𝐻))
mulgpropd.i	⊢ (𝜑 → 𝐵 ⊆ 𝐾)
mulgpropd.k	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐾)
mulgpropd.e	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)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.

Step	Hyp	Ref	Expression
1		mulgpropd.b1	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
2		mulgpropd.b2	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐻))
3		mulgpropdg.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
4		mulgpropdg.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
5		mulgpropd.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
6		ssel 3149	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐾 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐾))
7		ssel 3149	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐾 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐾))
8	6, 7	anim12d 335	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐾 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)))
9	5, 8	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)))
10	9	imp 124	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾))
11		mulgpropd.e	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
12	10, 11	syldan 282	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
13	1, 2, 3, 4, 12	grpidpropdg 12747	. . . . . 6 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
14	13	3ad2ant1 1018	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → (0g‘𝐺) = (0g‘𝐻))
15		1zzd 9278	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → 1 ∈ ℤ)
16		nnuz 9561	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
17	5	3ad2ant1 1018	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → 𝐵 ⊆ 𝐾)
18		simp3 999	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
19	17, 18	sseldd 3156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐾)
20	16, 19	ialgrlemconst 12037	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 ∈ (ℤ≥‘1)) → ((ℕ × {𝑏})‘𝑥) ∈ 𝐾)
21		mulgpropd.k	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐾)
22	21	3ad2antl1 1159	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐾)
23	11	3ad2antl1 1159	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
24	15, 20, 22, 23	seqfeq3 10509	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → seq1((+g‘𝐺), (ℕ × {𝑏})) = seq1((+g‘𝐻), (ℕ × {𝑏})))
25	24	fveq1d 5517	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎) = (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎))
26	1, 2, 3, 4, 12	grpinvpropdg 12899	. . . . . . . 8 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
27	26	3ad2ant1 1018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → (invg‘𝐺) = (invg‘𝐻))
28	24	fveq1d 5517	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → (seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎) = (seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))
29	27, 28	fveq12d 5522	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎)) = ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎)))
30	25, 29	ifeq12d 3553	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎))) = if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))))
31	14, 30	ifeq12d 3553	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎)))) = if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎)))))
32	31	mpoeq3dva 5938	. . 3 ⊢ (𝜑 → (𝑎 ∈ ℤ, 𝑏 ∈ 𝐵 ↦ if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎))))) = (𝑎 ∈ ℤ, 𝑏 ∈ 𝐵 ↦ if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))))))
33		eqidd 2178	. . . 4 ⊢ (𝜑 → ℤ = ℤ)
34		eqidd 2178	. . . 4 ⊢ (𝜑 → if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎)))) = if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎)))))
35	33, 1, 34	mpoeq123dv 5936	. . 3 ⊢ (𝜑 → (𝑎 ∈ ℤ, 𝑏 ∈ 𝐵 ↦ if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎))))) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎))))))
36		eqidd 2178	. . . 4 ⊢ (𝜑 → if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎)))) = if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎)))))
37	33, 2, 36	mpoeq123dv 5936	. . 3 ⊢ (𝜑 → (𝑎 ∈ ℤ, 𝑏 ∈ 𝐵 ↦ if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))))) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))))))
38	32, 35, 37	3eqtr3d 2218	. 2 ⊢ (𝜑 → (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎))))) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))))))
39		mulgpropdg.m	. . 3 ⊢ (𝜑 → · = (.g‘𝐺))
40		eqid 2177	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
41		eqid 2177	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
42		eqid 2177	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
43		eqid 2177	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
44		eqid 2177	. . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺)
45	40, 41, 42, 43, 44	mulgfvalg 12939	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (.g‘𝐺) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎))))))
46	3, 45	syl 14	. . 3 ⊢ (𝜑 → (.g‘𝐺) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎))))))
47	39, 46	eqtrd 2210	. 2 ⊢ (𝜑 → · = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐺) ↦ if(𝑎 = 0, (0g‘𝐺), if(0 < 𝑎, (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎))))))
48		mulgpropdg.n	. . 3 ⊢ (𝜑 → × = (.g‘𝐻))
49		eqid 2177	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
50		eqid 2177	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
51		eqid 2177	. . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻)
52		eqid 2177	. . . . 5 ⊢ (invg‘𝐻) = (invg‘𝐻)
53		eqid 2177	. . . . 5 ⊢ (.g‘𝐻) = (.g‘𝐻)
54	49, 50, 51, 52, 53	mulgfvalg 12939	. . . 4 ⊢ (𝐻 ∈ 𝑊 → (.g‘𝐻) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))))))
55	4, 54	syl 14	. . 3 ⊢ (𝜑 → (.g‘𝐻) = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))))))
56	48, 55	eqtrd 2210	. 2 ⊢ (𝜑 → × = (𝑎 ∈ ℤ, 𝑏 ∈ (Base‘𝐻) ↦ if(𝑎 = 0, (0g‘𝐻), if(0 < 𝑎, (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎), ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))))))
57	38, 47, 56	3eqtr4d 2220	1 ⊢ (𝜑 → · = × )

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  ℕcn 8917  ℤcz 9251  seqcseq 10442  Basecbs 12456  +_gcplusg 12530  0_gc0g 12695  inv_gcminusg 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