Theorem subgsub 13903

Description: The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)

Hypotheses

Ref	Expression
subgsubcl.p	⊢ − = (-g‘𝐺)
subgsub.h	⊢ 𝐻 = (𝐺 ↾s 𝑆)
subgsub.n	⊢ 𝑁 = (-g‘𝐻)

Assertion

Ref	Expression
subgsub	⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋𝑁𝑌))

Proof of Theorem subgsub

Step	Hyp	Ref	Expression
1		subgsub.h	. . . . . 6 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
2	1	a1i 9	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 = (𝐺 ↾s 𝑆))
3		eqidd 2233	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐺))
4		id 19	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺))
5		subgrcl 13896	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	2, 3, 4, 5	ressplusgd 13342	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (+g‘𝐺) = (+g‘𝐻))
7	6	3ad2ant1 1045	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (+g‘𝐺) = (+g‘𝐻))
8		eqidd 2233	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 = 𝑋)
9		eqid 2232	. . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺)
10		eqid 2232	. . . . 5 ⊢ (invg‘𝐻) = (invg‘𝐻)
11	1, 9, 10	subginv 13898	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑌 ∈ 𝑆) → ((invg‘𝐺)‘𝑌) = ((invg‘𝐻)‘𝑌))
12	11	3adant2 1043	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((invg‘𝐺)‘𝑌) = ((invg‘𝐻)‘𝑌))
13	7, 8, 12	oveq123d 6071	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = (𝑋(+g‘𝐻)((invg‘𝐻)‘𝑌)))
14		eqid 2232	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
15	14	subgss 13891	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
16	15	3ad2ant1 1045	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺))
17		simp2 1025	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ 𝑆)
18	16, 17	sseldd 3239	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺))
19		simp3 1026	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ 𝑆)
20	16, 19	sseldd 3239	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ (Base‘𝐺))
21		eqid 2232	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
22		subgsubcl.p	. . . 4 ⊢ − = (-g‘𝐺)
23	14, 21, 9, 22	grpsubval 13759	. . 3 ⊢ ((𝑋 ∈ (Base‘𝐺) ∧ 𝑌 ∈ (Base‘𝐺)) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)))
24	18, 20, 23	syl2anc 411	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)))
25	1	subgbas 13895	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
26	25	3ad2ant1 1045	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑆 = (Base‘𝐻))
27	17, 26	eleqtrd 2311	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐻))
28	19, 26	eleqtrd 2311	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ (Base‘𝐻))
29		eqid 2232	. . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻)
30		eqid 2232	. . . 4 ⊢ (+g‘𝐻) = (+g‘𝐻)
31		subgsub.n	. . . 4 ⊢ 𝑁 = (-g‘𝐻)
32	29, 30, 10, 31	grpsubval 13759	. . 3 ⊢ ((𝑋 ∈ (Base‘𝐻) ∧ 𝑌 ∈ (Base‘𝐻)) → (𝑋𝑁𝑌) = (𝑋(+g‘𝐻)((invg‘𝐻)‘𝑌)))
33	27, 28, 32	syl2anc 411	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋𝑁𝑌) = (𝑋(+g‘𝐻)((invg‘𝐻)‘𝑌)))
34	13, 24, 33	3eqtr4d 2275	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋𝑁𝑌))

Syntax hints:   → wi 4   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ⊆ wss 3211  ‘cfv 5352  (class class class)co 6050  Basecbs 13212   ↾_s cress 13213  +_gcplusg 13290  Grpcgrp 13713  inv_gcminusg 13714  -_gcsg 13715  SubGrpcsubg 13884

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 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  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 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-sbg 13718  df-subg 13887

This theorem is referenced by:  zringsubgval  14753  zndvds  14797