isnghm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > isnghm		Structured version Visualization version GIF version

Theorem isnghm 22753

Description: A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)

**Hypothesis**
Ref	Expression
nmofval.1	⊢ 𝑁 = (𝑆 normOp 𝑇)

**Assertion**
Ref	Expression
isnghm	⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ)))

**Proof of Theorem isnghm**
Step	Hyp	Ref	Expression
1		nmofval.1	. . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇)
2	1	nghmfval 22752	. . 3 ⊢ (𝑆 NGHom 𝑇) = (^◡𝑁 “ ℝ)
3	2	eleq2i 2840	. 2 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ 𝐹 ∈ (^◡𝑁 “ ℝ))
4		n0i 4065	. . . 4 ⊢ (𝐹 ∈ (^◡𝑁 “ ℝ) → ¬ (^◡𝑁 “ ℝ) = ∅)
5		nmoffn 22741	. . . . . . . . . . 11 ⊢ normOp Fn (NrmGrp × NrmGrp)
6		fndm 6129	. . . . . . . . . . 11 ⊢ ( normOp Fn (NrmGrp × NrmGrp) → dom normOp = (NrmGrp × NrmGrp))
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ dom normOp = (NrmGrp × NrmGrp)
8	7	ndmov 6963	. . . . . . . . 9 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅)
9	1, 8	syl5eq 2815	. . . . . . . 8 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅)
10	9	cnveqd 5435	. . . . . . 7 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ^◡𝑁 = ^◡∅)
11		cnv0 5675	. . . . . . 7 ⊢ ^◡∅ = ∅
12	10, 11	syl6eq 2819	. . . . . 6 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ^◡𝑁 = ∅)
13	12	imaeq1d 5605	. . . . 5 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (^◡𝑁 “ ℝ) = (∅ “ ℝ))
14		0ima 5622	. . . . 5 ⊢ (∅ “ ℝ) = ∅
15	13, 14	syl6eq 2819	. . . 4 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (^◡𝑁 “ ℝ) = ∅)
16	4, 15	nsyl2 144	. . 3 ⊢ (𝐹 ∈ (^◡𝑁 “ ℝ) → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp))
17	1	nmof 22749	. . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁:(𝑆 GrpHom 𝑇)⟶ℝ^*)
18		ffn 6184	. . . 4 ⊢ (𝑁:(𝑆 GrpHom 𝑇)⟶ℝ^* → 𝑁 Fn (𝑆 GrpHom 𝑇))
19		elpreima 6479	. . . 4 ⊢ (𝑁 Fn (𝑆 GrpHom 𝑇) → (𝐹 ∈ (^◡𝑁 “ ℝ) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ)))
20	17, 18, 19	3syl 18	. . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝐹 ∈ (^◡𝑁 “ ℝ) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ)))
21	16, 20	biadan2 674	. 2 ⊢ (𝐹 ∈ (^◡𝑁 “ ℝ) ↔ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ)))
22	3, 21	bitri 264	1 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 383 = wceq 1629 ∈ wcel 2143 ∅c0 4060 × cxp 5246 ^◡ccnv 5247 dom cdm 5248 “ cima 5251 Fn wfn 6025 ⟶wf 6026 ‘cfv 6030 (class class class)co 6791 ℝcr 10135 ℝ^*cxr 10273 GrpHom cghm 17871 NrmGrpcngp 22608 normOp cnmo 22735 NGHom cnghm 22736

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1868 ax-4 1883 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2145 ax-9 2152 ax-10 2172 ax-11 2188 ax-12 2201 ax-13 2406 ax-ext 2749 ax-sep 4911 ax-nul 4919 ax-pow 4970 ax-pr 5033 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1070 df-3an 1071 df-tru 1632 df-ex 1851 df-nf 1856 df-sb 2048 df-eu 2620 df-mo 2621 df-clab 2756 df-cleq 2762 df-clel 2765 df-nfc 2900 df-ne 2942 df-nel 3045 df-ral 3064 df-rex 3065 df-reu 3066 df-rmo 3067 df-rab 3068 df-v 3350 df-sbc 3585 df-csb 3680 df-dif 3723 df-un 3725 df-in 3727 df-ss 3734 df-nul 4061 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4572 df-iun 4653 df-br 4784 df-opab 4844 df-mpt 4861 df-id 5156 df-po 5169 df-so 5170 df-xp 5254 df-rel 5255 df-cnv 5256 df-co 5257 df-dm 5258 df-rn 5259 df-res 5260 df-ima 5261 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-1st 7313 df-2nd 7314 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-sup 8502 df-inf 8503 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-ico 12385 df-nmo 22738 df-nghm 22739

This theorem is referenced by: isnghm2 22754 nghmcl 22757 nmoi 22758 nghmrcl1 22762 nghmrcl2 22763 nghmghm 22764 isnmhm2 22782

W3C validator