Theorem isnsg2 18300

Description: Weaken the condition of isnsg 18299 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
isnsg.1	⊢ 𝑋 = (Base‘𝐺)
isnsg.2	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
isnsg2	⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥, + ,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Proof of Theorem isnsg2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnsg.1	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		isnsg.2	. . 3 ⊢ + = (+g‘𝐺)
3	1, 2	isnsg 18299	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆)))
4		dfbi2 478	. . . . . . 7 ⊢ (((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ (((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ∧ ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆)))
5	4	ralbii 3133	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 (((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ∧ ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆)))
6	5	ralbii 3133	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ∧ ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆)))
7		r19.26-2 3138	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ∧ ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆)) ↔ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆)))
8	6, 7	bitri 278	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆)))
9		oveq2 7143	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑥 + 𝑧) = (𝑥 + 𝑦))
10	9	eleq1d 2874	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑥 + 𝑦) ∈ 𝑆))
11		oveq1 7142	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑥) = (𝑦 + 𝑥))
12	11	eleq1d 2874	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑧 + 𝑥) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
13	10, 12	imbi12d 348	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ↔ ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
14	13	cbvralvw 3396	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
15	14	ralbii 3133	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
16		ralcom 3307	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆))
17		oveq2 7143	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑧 + 𝑥) = (𝑧 + 𝑦))
18	17	eleq1d 2874	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑧 + 𝑥) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆))
19		oveq1 7142	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑧) = (𝑦 + 𝑧))
20	19	eleq1d 2874	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑦 + 𝑧) ∈ 𝑆))
21	18, 20	imbi12d 348	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆)))
22	21	cbvralvw 3396	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆))
23	22	ralbii 3133	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆))
24		oveq1 7142	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
25	24	eleq1d 2874	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑧 + 𝑦) ∈ 𝑆 ↔ (𝑥 + 𝑦) ∈ 𝑆))
26		oveq2 7143	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑦 + 𝑧) = (𝑦 + 𝑥))
27	26	eleq1d 2874	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
28	25, 27	imbi12d 348	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆) ↔ ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
29	28	ralbidv 3162	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
30	29	cbvralvw 3396	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
31	16, 23, 30	3bitri 300	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
32	15, 31	anbi12i 629	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆)) ↔ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
33		anidm 568	. . . 4 ⊢ ((∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
34	8, 32, 33	3bitri 300	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
35	34	anbi2i 625	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆)) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
36	3, 35	bitri 278	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  SubGrpcsubg 18265  NrmSGrpcnsg 18266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-subg 18268  df-nsg 18269

This theorem is referenced by:  isnsg3  18304  tgpconncomp  22718