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Theorem sspg 30703

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 2731	. . . . . . . . . . 11 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		sspg.g	. . . . . . . . . . 11 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
3	1, 2	nvgf 30593	. . . . . . . . . 10 ⊢ (𝑈 ∈ NrmCVec → 𝐺:((BaseSet‘𝑈) × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4	3	ffund 6655	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → Fun 𝐺)
5	4	funresd 6524	. . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec → Fun (𝐺 ↾ (𝑌 × 𝑌)))
6	5	adantr 480	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → Fun (𝐺 ↾ (𝑌 × 𝑌)))
7		sspg.h	. . . . . . . . . 10 ⊢ 𝐻 = (SubSp‘𝑈)
8	7	sspnv 30701	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ NrmCVec)
9		sspg.y	. . . . . . . . . 10 ⊢ 𝑌 = (BaseSet‘𝑊)
10		sspg.f	. . . . . . . . . 10 ⊢ 𝐹 = ( +_𝑣 ‘𝑊)
11	9, 10	nvgf 30593	. . . . . . . . 9 ⊢ (𝑊 ∈ NrmCVec → 𝐹:(𝑌 × 𝑌)⟶𝑌)
12	8, 11	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹:(𝑌 × 𝑌)⟶𝑌)
13	12	ffnd 6652	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 Fn (𝑌 × 𝑌))
14		fnresdm 6600	. . . . . . . . 9 ⊢ (𝐹 Fn (𝑌 × 𝑌) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) = 𝐹)
16		eqid 2731	. . . . . . . . . . . 12 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
17		eqid 2731	. . . . . . . . . . . 12 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
18		eqid 2731	. . . . . . . . . . . 12 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
19		eqid 2731	. . . . . . . . . . . 12 ⊢ (norm_CV‘𝑊) = (norm_CV‘𝑊)
20	2, 10, 16, 17, 18, 19, 7	isssp 30699	. . . . . . . . . . 11 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ ( ·_𝑠OLD ‘𝑊) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑊) ⊆ (norm_CV‘𝑈)))))
21	20	simplbda 499	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐹 ⊆ 𝐺 ∧ ( ·_𝑠OLD ‘𝑊) ⊆ ( ·_𝑠OLD ‘𝑈) ∧ (norm_CV‘𝑊) ⊆ (norm_CV‘𝑈)))
22	21	simp1d 1142	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 ⊆ 𝐺)
23		ssres 5952	. . . . . . . . 9 ⊢ (𝐹 ⊆ 𝐺 → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
24	22, 23	syl 17	. . . . . . . 8 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐹 ↾ (𝑌 × 𝑌)) ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
25	15, 24	eqsstrrd 3970	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌)))
26	6, 13, 25	3jca 1128	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))))
27		oprssov 7515	. . . . . 6 ⊢ (((Fun (𝐺 ↾ (𝑌 × 𝑌)) ∧ 𝐹 Fn (𝑌 × 𝑌) ∧ 𝐹 ⊆ (𝐺 ↾ (𝑌 × 𝑌))) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
28	26, 27	sylan 580	. . . . 5 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦) = (𝑥𝐹𝑦))
29	28	eqcomd 2737	. . . 4 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌)) → (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
30	29	ralrimivva 3175	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))
31		eqid 2731	. . 3 ⊢ (𝑌 × 𝑌) = (𝑌 × 𝑌)
32	30, 31	jctil 519	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦)))
33	3	ffnd 6652	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
34	33	adantr 480	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
35	1, 9, 7	sspba 30702	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ (BaseSet‘𝑈))
36		xpss12 5631	. . . . 5 ⊢ ((𝑌 ⊆ (BaseSet‘𝑈) ∧ 𝑌 ⊆ (BaseSet‘𝑈)) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
37	35, 35, 36	syl2anc 584	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈)))
38		fnssres 6604	. . . 4 ⊢ ((𝐺 Fn ((BaseSet‘𝑈) × (BaseSet‘𝑈)) ∧ (𝑌 × 𝑌) ⊆ ((BaseSet‘𝑈) × (BaseSet‘𝑈))) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
39	34, 37, 38	syl2anc 584	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐺 ↾ (𝑌 × 𝑌)) Fn (𝑌 × 𝑌))
40		eqfnov 7475	. . 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))))
41	13, 39, 40	syl2anc 584	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐹 = (𝐺 ↾ (𝑌 × 𝑌)) ↔ ((𝑌 × 𝑌) = (𝑌 × 𝑌) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑥𝐹𝑦) = (𝑥(𝐺 ↾ (𝑌 × 𝑌))𝑦))))
42	32, 41	mpbird 257	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ⊆ wss 3902 × cxp 5614 ↾ cres 5618 Fun wfun 6475 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 NrmCVeccnv 30559 +_𝑣 cpv 30560 BaseSetcba 30561 ·_𝑠OLD cns 30562 norm_CVcnmcv 30565 SubSpcss 30696

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-1st 7921 df-2nd 7922 df-grpo 30468 df-ablo 30520 df-vc 30534 df-nv 30567 df-va 30570 df-ba 30571 df-sm 30572 df-0v 30573 df-nmcv 30575 df-ssp 30697

This theorem is referenced by: sspgval 30704

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