Theorem sspg 27583

Description: Vector addition on a subspace is a restriction of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
sspg.y	⊢ 𝑌 = (BaseSet‘𝑊)
sspg.g	⊢ 𝐺 = ( +𝑣 ‘𝑈)
sspg.f	⊢ 𝐹 = ( +𝑣 ‘𝑊)
sspg.h	⊢ 𝐻 = (SubSp‘𝑈)

Assertion

Ref	Expression
sspg	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))

Proof of Theorem sspg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . . . 11 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		sspg.g	. . . . . . . . . . 11 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
3	1, 2	nvgf 27473	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4		ffun 6048	. . . . . . . . . 10 ⊢ (𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) → Fun 𝐺)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → Fun 𝐺)
6		funres 5929	. . . . . . . . 9 ⊢ (Fun 𝐺 → Fun (𝐺 ↾ (𝑌 × 𝑌)))
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec → Fun (𝐺 ↾ (𝑌 × 𝑌)))
8	7	adantr 481	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → Fun (𝐺 ↾ (𝑌 × 𝑌)))
9		sspg.h	. . . . . . . . . 10 ⊢ 𝐻 = (SubSp‘𝑈)
10	9	sspnv 27581	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)
11		sspg.y	. . . . . . . . . 10 ⊢ 𝑌 = (BaseSet‘𝑊)
12		sspg.f	. . . . . . . . . 10 ⊢ 𝐹 = ( +𝑣 ‘𝑊)
13	11, 12	nvgf 27473	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑌)
14	10, 13	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹:(𝑌 × 𝑌)⟶𝑌)
15		ffn 6045	. . . . . . . 8 ⊢ (𝐹:(𝑌 × 𝑌)⟶𝑌 → 𝐹 Fn (𝑌 × 𝑌))
16	14, 15	syl 17	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 Fn (𝑌 × 𝑌))
17		fnresdm 6000	. . . . . . . . 9 ⊢ (𝐹 Fn (𝑌 × 𝑌) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
18	16, 17	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
19		eqid 2622	. . . . . . . . . . . 12 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
20		eqid 2622	. . . . . . . . . . . 12 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊)
21		eqid 2622	. . . . . . . . . . . 12 ⊢ (normCV‘𝑈) = (normCV‘𝑈)
22		eqid 2622	. . . . . . . . . . . 12 ⊢ (normCV‘𝑊) = (normCV‘𝑊)
23	2, 12, 19, 20, 21, 22, 9	isssp 27579	. . . . . . . . . . 11 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))))
24	23	simplbda 654	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐹 ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈)))
25	24	simp1d 1073	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 ⊆ 𝐺)
26		ssres 5424	. . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
28	18, 27	eqsstr3d 3640	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
29	8, 16, 28	3jca 1242	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))))
30		oprssov 6803	. . . . . 6 ⊢ (((Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
31	29, 30	sylan 488	. . . . 5 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
32	31	eqcomd 2628	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
33	32	ralrimivva 2971	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
34		eqid 2622	. . 3 ⊢ (𝑌 × 𝑌) = (𝑌 × 𝑌)
35	33, 34	jctil 560	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
36		ffn 6045	. . . . . 6 ⊢ (𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
37	3, 36	syl 17	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
38	37	adantr 481	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
39	1, 11, 9	sspba 27582	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
40		xpss12 5225	. . . . 5 ⊢ ((𝑌 ⊆ (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
41	39, 39, 40	syl2anc 693	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
42		fnssres 6004	. . . 4 ⊢ ((𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)) ∧ (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈))) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
43	38, 41, 42	syl2anc 693	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
44		eqfnov 6766	. . 3 ⊢ ((𝐹 Fn (𝑌 × 𝑌) ∧ (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌)) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
45	16, 43, 44	syl2anc 693	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
46	35, 45	mpbird 247	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574   × cxp 5112   ↾ cres 5116  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  NrmCVeccnv 27439   +_𝑣 cpv 27440  BaseSetcba 27441   ·_𝑠OLD cns 27442  norm_CVcnmcv 27445  SubSpcss 27576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-grpo 27347  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-ssp 27577

This theorem is referenced by:  sspgval  27584