Theorem nghmfval 22431

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
nghmfval	⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)

Proof of Theorem nghmfval

Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq12 6614	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = (𝑆 normOp 𝑇))
2		nmofval.1	. . . . . 6 ⊢ 𝑁 = (𝑆 normOp 𝑇)
3	1, 2	syl6eqr 2678	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁)
4	3	cnveqd 5263	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ◡(𝑠 normOp 𝑡) = ◡𝑁)
5	4	imaeq1d 5428	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (◡(𝑠 normOp 𝑡) “ ℝ) = (◡𝑁 “ ℝ))
6		df-nghm 22418	. . 3 ⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ))
7	2	ovexi 6634	. . . . 5 ⊢ 𝑁 ∈ V
8	7	cnvex 7063	. . . 4 ⊢ ◡𝑁 ∈ V
9	8	imaex 7052	. . 3 ⊢ (◡𝑁 “ ℝ) ∈ V
10	5, 6, 9	ovmpt2a 6745	. 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ))
11	6	mpt2ndm0 6829	. . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = ∅)
12		nmoffn 22420	. . . . . . . . . 10 ⊢ normOp Fn (NrmGrp × NrmGrp)
13		fndm 5950	. . . . . . . . . 10 ⊢ ( normOp Fn (NrmGrp × NrmGrp) → dom normOp = (NrmGrp × NrmGrp))
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ dom normOp = (NrmGrp × NrmGrp)
15	14	ndmov 6772	. . . . . . . 8 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅)
16	2, 15	syl5eq 2672	. . . . . . 7 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅)
17	16	cnveqd 5263	. . . . . 6 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ◡∅)
18		cnv0 5498	. . . . . 6 ⊢ ◡∅ = ∅
19	17, 18	syl6eq 2676	. . . . 5 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ∅)
20	19	imaeq1d 5428	. . . 4 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = (∅ “ ℝ))
21		0ima 5445	. . . 4 ⊢ (∅ “ ℝ) = ∅
22	20, 21	syl6eq 2676	. . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = ∅)
23	11, 22	eqtr4d 2663	. 2 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ))
24	10, 23	pm2.61i 176	1 ⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1480   ∈ wcel 1992  ∅c0 3896   × cxp 5077  ^◡ccnv 5078  dom cdm 5079   “ cima 5082   Fn wfn 5845  (class class class)co 6605  ℝcr 9880  NrmGrpcngp 22287   normOp cnmo 22414   NGHom cnghm 22415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-sup 8293  df-inf 8294  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-ico 12120  df-nmo 22417  df-nghm 22418

This theorem is referenced by:  isnghm  22432