Theorem submmulg 13872

Description: A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)

Hypotheses

Ref	Expression
submmulgcl.t	⊢ ∙ = (.g‘𝐺)
submmulg.h	⊢ 𝐻 = (𝐺 ↾s 𝑆)
submmulg.t	⊢ · = (.g‘𝐻)

Assertion

Ref	Expression
submmulg	⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) = (𝑁 · 𝑋))

Proof of Theorem submmulg

Step	Hyp	Ref	Expression
1		simpl1 1027	. . . . . 6 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ (SubMnd‘𝐺))
2		submmulg.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
3	2	a1i 9	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐻 = (𝐺 ↾s 𝑆))
4		eqidd 2233	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (+g‘𝐺) = (+g‘𝐺))
5		id 19	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
6		submrcl 13673	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
7	3, 4, 5, 6	ressplusgd 13331	. . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
8	1, 7	syl 14	. . . . 5 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (+g‘𝐺) = (+g‘𝐻))
9	8	seqeq2d 10812	. . . 4 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})))
10	9	fveq1d 5671	. . 3 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁) = (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁))
11		simpr 110	. . . 4 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
12		eqid 2232	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
13	12	submss 13678	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
14	13	3ad2ant1 1045	. . . . . 6 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺))
15		simp3 1026	. . . . . 6 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
16	14, 15	sseldd 3238	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺))
17	16	adantr 276	. . . 4 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ (Base‘𝐺))
18		eqid 2232	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
19		submmulgcl.t	. . . . 5 ⊢ ∙ = (.g‘𝐺)
20		eqid 2232	. . . . 5 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))
21	12, 18, 19, 20	mulgnn 13832	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (𝑁 ∙ 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁))
22	11, 17, 21	syl2anc 411	. . 3 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (𝑁 ∙ 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁))
23	2	submbas 13683	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 = (Base‘𝐻))
24	23	3ad2ant1 1045	. . . . . 6 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑆 = (Base‘𝐻))
25	15, 24	eleqtrd 2311	. . . . 5 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻))
26	25	adantr 276	. . . 4 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ (Base‘𝐻))
27		eqid 2232	. . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻)
28		eqid 2232	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
29		submmulg.t	. . . . 5 ⊢ · = (.g‘𝐻)
30		eqid 2232	. . . . 5 ⊢ seq1((+g‘𝐻), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋}))
31	27, 28, 29, 30	mulgnn 13832	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑁 · 𝑋) = (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁))
32	11, 26, 31	syl2anc 411	. . 3 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (𝑁 · 𝑋) = (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁))
33	10, 22, 32	3eqtr4d 2275	. 2 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 ∈ ℕ) → (𝑁 ∙ 𝑋) = (𝑁 · 𝑋))
34		simpl1 1027	. . . . 5 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → 𝑆 ∈ (SubMnd‘𝐺))
35		eqid 2232	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
36	2, 35	subm0 13684	. . . . 5 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (0g‘𝐺) = (0g‘𝐻))
37	34, 36	syl 14	. . . 4 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (0g‘𝐺) = (0g‘𝐻))
38	16	adantr 276	. . . . 5 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → 𝑋 ∈ (Base‘𝐺))
39	12, 35, 19	mulg0 13831	. . . . 5 ⊢ (𝑋 ∈ (Base‘𝐺) → (0 ∙ 𝑋) = (0g‘𝐺))
40	38, 39	syl 14	. . . 4 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (0 ∙ 𝑋) = (0g‘𝐺))
41	25	adantr 276	. . . . 5 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → 𝑋 ∈ (Base‘𝐻))
42		eqid 2232	. . . . . 6 ⊢ (0g‘𝐻) = (0g‘𝐻)
43	27, 42, 29	mulg0 13831	. . . . 5 ⊢ (𝑋 ∈ (Base‘𝐻) → (0 · 𝑋) = (0g‘𝐻))
44	41, 43	syl 14	. . . 4 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (0 · 𝑋) = (0g‘𝐻))
45	37, 40, 44	3eqtr4d 2275	. . 3 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (0 ∙ 𝑋) = (0 · 𝑋))
46		simpr 110	. . . 4 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → 𝑁 = 0)
47	46	oveq1d 6064	. . 3 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (𝑁 ∙ 𝑋) = (0 ∙ 𝑋))
48	46	oveq1d 6064	. . 3 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (𝑁 · 𝑋) = (0 · 𝑋))
49	45, 47, 48	3eqtr4d 2275	. 2 ⊢ (((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) ∧ 𝑁 = 0) → (𝑁 ∙ 𝑋) = (𝑁 · 𝑋))
50		simp2 1025	. . 3 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → 𝑁 ∈ ℕ0)
51		elnn0 9494	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
52	50, 51	sylib 122	. 2 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
53	33, 49, 52	mpjaodan 806	1 ⊢ ((𝑆 ∈ (SubMnd‘𝐺) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑆) → (𝑁 ∙ 𝑋) = (𝑁 · 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ⊆ wss 3210  {csn 3688   × cxp 4746  ‘cfv 5351  (class class class)co 6049  0cc0 8123  1c1 8124  ℕcn 9233  ℕ₀cn0 9492  seqcseq 10805  Basecbs 13201   ↾_s cress 13202  +_gcplusg 13279  0_gc0g 13458  Mndcmnd 13618  SubMndcsubmnd 13660  ._gcmg 13825

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-2 9292  df-n0 9493  df-z 9574  df-uz 9850  df-seqfrec 10806  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-0g 13460  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-submnd 13662  df-minusg 13706  df-mulg 13826

This theorem is referenced by:  lgseisenlem4  15933