Theorem isnsg2 19187

Description: Weaken the condition of isnsg 19186 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 19186	. 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 3107	. . . . . 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 3107	. . . . 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 3146	. . . . 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 277	. . . 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 7398	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑥 + 𝑧) = (𝑥 + 𝑦))
10	9	eleq1d 2846	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑥 + 𝑦) ∈ 𝑆))
11		oveq1 7397	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑥) = (𝑦 + 𝑥))
12	11	eleq1d 2846	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑧 + 𝑥) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
13	10, 12	imbi12d 346	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ↔ ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
14	13	cbvralvw 3239	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
15	14	ralbii 3107	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 → (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
16		ralcom 3289	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆))
17		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑧 + 𝑥) = (𝑧 + 𝑦))
18	17	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑧 + 𝑥) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆))
19		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑧) = (𝑦 + 𝑧))
20	19	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑦 + 𝑧) ∈ 𝑆))
21	18, 20	imbi12d 346	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆)))
22	21	cbvralvw 3239	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆))
23	22	ralbii 3107	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑧 + 𝑥) ∈ 𝑆 → (𝑥 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆))
24		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
25	24	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑧 + 𝑦) ∈ 𝑆 ↔ (𝑥 + 𝑦) ∈ 𝑆))
26		oveq2 7398	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑦 + 𝑧) = (𝑦 + 𝑥))
27	26	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))
28	25, 27	imbi12d 346	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆) ↔ ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
29	28	ralbidv 3184	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝑋 ((𝑧 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑧) ∈ 𝑆) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))
30	29	cbvralvw 3239	. . . . . 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 299	. . . . 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 637	. . . 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 572	. . . 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 299	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑥) ∈ 𝑆) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆))
35	34	anbi2i 632	. 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 277	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 → (𝑦 + 𝑥) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ‘cfv 6515  (class class class)co 7390  Basecbs 17235  +_gcplusg 17276  SubGrpcsubg 19152  NrmSGrpcnsg 19153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fv 6523  df-ov 7393  df-subg 19155  df-nsg 19156

This theorem is referenced by:  isnsg3  19191  subrngringnsg  20589  tgpconncomp  24160  opprnsg  33632