Theorem isnsg2 19036

Description: Weaken the condition of isnsg 19035 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 19035	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆)))
4		dfbi2 476	. . . . . . 7 ⊢ (((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ (((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ∧ ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆)))
5	4	ralbii 3094	. . . . . 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 3094	. . . . 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 3139	. . . . 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 275	. . . 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 7417	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑥 + 𝑧) = (𝑥 + 𝑦))
10	9	eleq1d 2819	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑥 + 𝑦) ∈ 𝑆))
11		oveq1 7416	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑥) = (𝑦 + 𝑥))
12	11	eleq1d 2819	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑧 + 𝑥) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
13	10, 12	imbi12d 345	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ↔ ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
14	13	cbvralvw 3235	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
15	14	ralbii 3094	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
16		ralcom 3287	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆))
17		oveq2 7417	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑧 + 𝑥) = (𝑧 + 𝑦))
18	17	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑧 + 𝑥) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆))
19		oveq1 7416	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑧) = (𝑦 + 𝑧))
20	19	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑦 + 𝑧) ∈ 𝑆))
21	18, 20	imbi12d 345	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆)))
22	21	cbvralvw 3235	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆))
23	22	ralbii 3094	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆))
24		oveq1 7416	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
25	24	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑧 + 𝑦) ∈ 𝑆 ↔ (𝑥 + 𝑦) ∈ 𝑆))
26		oveq2 7417	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑦 + 𝑧) = (𝑦 + 𝑥))
27	26	eleq1d 2819	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
28	25, 27	imbi12d 345	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆) ↔ ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
29	28	ralbidv 3178	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
30	29	cbvralvw 3235	. . . . . 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 297	. . . . 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 628	. . . 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 566	. . . 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 297	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
35	34	anbi2i 624	. 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 275	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  SubGrpcsubg 19000  NrmSGrpcnsg 19001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-subg 19003  df-nsg 19004

This theorem is referenced by:  isnsg3  19040  tgpconncomp  23617  opprnsg  32598  subrngringnsg  46732