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Theorem isnghm 22747

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 22746	. . 3 ⊢ (𝑆 NGHom 𝑇) = (^◡𝑁 “ ℝ)
3	2	eleq2i 2842	. 2 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ 𝐹 ∈ (^◡𝑁 “ ℝ))
4		n0i 4068	. . . 4 ⊢ (𝐹 ∈ (^◡𝑁 “ ℝ) → ¬ (^◡𝑁 “ ℝ) = ∅)
5		nmoffn 22735	. . . . . . . . . . 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 6969	. . . . . . . . 9 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅)
9	1, 8	syl5eq 2817	. . . . . . . 8 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅)
10	9	cnveqd 5435	. . . . . . 7 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ^◡𝑁 = ^◡∅)
11		cnv0 5675	. . . . . . 7 ⊢ ^◡∅ = ∅
12	10, 11	syl6eq 2821	. . . . . 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 2821	. . . 4 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (^◡𝑁 “ ℝ) = ∅)
16	4, 15	nsyl2 144	. . 3 ⊢ (𝐹 ∈ (^◡𝑁 “ ℝ) → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp))
17	1	nmof 22743	. . . 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 6482	. . . 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 820	. 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 382 = wceq 1631 ∈ wcel 2145 ∅c0 4063 × cxp 5248 ^◡ccnv 5249 dom cdm 5250 “ cima 5253 Fn wfn 6025 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 ℝcr 10141 ℝ^*cxr 10279 GrpHom cghm 17865 NrmGrpcngp 22602 normOp cnmo 22729 NGHom cnghm 22730

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-po 5171 df-so 5172 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 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 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-1st 7319 df-2nd 7320 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-sup 8508 df-inf 8509 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-ico 12386 df-nmo 22732 df-nghm 22733

This theorem is referenced by: isnghm2 22748 nghmcl 22751 nmoi 22752 nghmrcl1 22756 nghmrcl2 22757 nghmghm 22758 isnmhm2 22776

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