Theorem mulgpropd 18269

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	⊢ · = (.g‘𝐺)
mulgpropd.n	⊢ × = (.g‘𝐻)
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
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.

Step	Hyp	Ref	Expression
1		mulgpropd.b1	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
2		mulgpropd.b2	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐻))
3		mulgpropd.i	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐾)
4		ssel 3961	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐾 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐾))
5		ssel 3961	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ 𝐾 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐾))
6	4, 5	anim12d 610	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐾 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)))
7	3, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)))
8	7	imp 409	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾))
9		mulgpropd.e	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
10	8, 9	syldan 593	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
11	1, 2, 10	grpidpropd 17872	. . . . . 6 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
12	11	3ad2ant1 1129	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → (0g‘𝐺) = (0g‘𝐻))
13		1zzd 12014	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → 1 ∈ ℤ)
14		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑏 ∈ V
15	14	fvconst2 6966	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((ℕ × {𝑏})‘𝑥) = 𝑏)
16		nnuz 12282	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ≥‘1)
17	16	eqcomi 2830	. . . . . . . . . . 11 ⊢ (ℤ≥‘1) = ℕ
18	15, 17	eleq2s 2931	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℤ≥‘1) → ((ℕ × {𝑏})‘𝑥) = 𝑏)
19	18	adantl 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 ∈ (ℤ≥‘1)) → ((ℕ × {𝑏})‘𝑥) = 𝑏)
20	3	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → 𝐵 ⊆ 𝐾)
21		simp3 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
22	20, 21	sseldd 3968	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐾)
23	22	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 ∈ (ℤ≥‘1)) → 𝑏 ∈ 𝐾)
24	19, 23	eqeltrd 2913	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) ∧ 𝑥 ∈ (ℤ≥‘1)) → ((ℕ × {𝑏})‘𝑥) ∈ 𝐾)
25		mulgpropd.k	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐾)
26	25	3ad2antl1 1181	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) ∈ 𝐾)
27	9	3ad2antl1 1181	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝐻)𝑦))
28	13, 24, 26, 27	seqfeq3 13421	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → seq1((+g‘𝐺), (ℕ × {𝑏})) = seq1((+g‘𝐻), (ℕ × {𝑏})))
29	28	fveq1d 6672	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → (seq1((+g‘𝐺), (ℕ × {𝑏}))‘𝑎) = (seq1((+g‘𝐻), (ℕ × {𝑏}))‘𝑎))
30	1, 2, 10	grpinvpropd 18174	. . . . . . . 8 ⊢ (𝜑 → (invg‘𝐺) = (invg‘𝐻))
31	30	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → (invg‘𝐺) = (invg‘𝐻))
32	28	fveq1d 6672	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → (seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎) = (seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎))
33	31, 32	fveq12d 6677	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ 𝐵) → ((invg‘𝐺)‘(seq1((+g‘𝐺), (ℕ × {𝑏}))‘-𝑎)) = ((invg‘𝐻)‘(seq1((+g‘𝐻), (ℕ × {𝑏}))‘-𝑎)))
34	29, 33	ifeq12d 4487	. . . . 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))))
35	12, 34	ifeq12d 4487	. . . 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)))))
36	35	mpoeq3dva 7231	. . 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))))))
37		eqidd 2822	. . . 4 ⊢ (𝜑 → ℤ = ℤ)
38		eqidd 2822	. . . 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)))))
39	37, 1, 38	mpoeq123dv 7229	. . 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))))))
40		eqidd 2822	. . . 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)))))
41	37, 2, 40	mpoeq123dv 7229	. . 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))))))
42	36, 39, 41	3eqtr3d 2864	. 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))))))
43		eqid 2821	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
44		eqid 2821	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
45		eqid 2821	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
46		eqid 2821	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
47		mulgpropd.m	. . 3 ⊢ · = (.g‘𝐺)
48	43, 44, 45, 46, 47	mulgfval 18226	. 2 ⊢ · = (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)))))
49		eqid 2821	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
50		eqid 2821	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
51		eqid 2821	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
52		eqid 2821	. . 3 ⊢ (invg‘𝐻) = (invg‘𝐻)
53		mulgpropd.n	. . 3 ⊢ × = (.g‘𝐻)
54	49, 50, 51, 52, 53	mulgfval 18226	. 2 ⊢ × = (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	42, 48, 54	3eqtr4g 2881	1 ⊢ (𝜑 → · = × )

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936  ifcif 4467  {csn 4567   class class class wbr 5066   × cxp 5553  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  0cc0 10537  1c1 10538   < clt 10675  -cneg 10871  ℕcn 11638  ℤcz 11982  ℤ_≥cuz 12244  seqcseq 13370  Basecbs 16483  +_gcplusg 16565  0_gc0g 16713  inv_gcminusg 18104  ._gcmg 18224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-seq 13371  df-0g 16715  df-minusg 18107  df-mulg 18225

This theorem is referenced by:  mulgass3  19387  coe1tm  20441  ply1coe  20464  evl1expd  20508