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Theorem pjnormssi 32188

Description: Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
pjco.1	⊢ 𝐺 ∈ C_ℋ
pjco.2	⊢ 𝐻 ∈ C_ℋ

**Assertion**
Ref	Expression
pjnormssi	⊢ (𝐺 ⊆ 𝐻 ↔ ∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)))

Distinct variable groups: 𝑥,𝐻 𝑥,𝐺

**Proof of Theorem pjnormssi**
Step	Hyp	Ref	Expression
1		pjco.2	. . . . . . 7 ⊢ 𝐻 ∈ C_ℋ
2		pjco.1	. . . . . . 7 ⊢ 𝐺 ∈ C_ℋ
3	1, 2	pjssmi 32185	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ 𝐻 → (((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) = ((proj_ℎ‘(𝐻 ∩ (⊥‘𝐺)))‘𝑥)))
4	1, 2	pjssge0i 32186	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) = ((proj_ℎ‘(𝐻 ∩ (⊥‘𝐺)))‘𝑥) → 0 ≤ ((((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) ·_ih 𝑥)))
5	3, 4	syld 47	. . . . 5 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ 𝐻 → 0 ≤ ((((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) ·_ih 𝑥)))
6	1, 2	pjdifnormi 32187	. . . . 5 ⊢ (𝑥 ∈ ℋ → (0 ≤ ((((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) ·_ih 𝑥) ↔ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))))
7	5, 6	sylibd 239	. . . 4 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ 𝐻 → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))))
8	7	com12 32	. . 3 ⊢ (𝐺 ⊆ 𝐻 → (𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))))
9	8	ralrimiv 3144	. 2 ⊢ (𝐺 ⊆ 𝐻 → ∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)))
10	1	choccli 31327	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ C_ℋ
11	10	cheli 31252	. . . . . . 7 ⊢ (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ ℋ)
12		breq2 5146	. . . . . . . . . . . . 13 ⊢ ((norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0 → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) ↔ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0))
13	12	biimpac 478	. . . . . . . . . . . 12 ⊢ (((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) ∧ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0)
14	2	pjhcli 31438	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ)
15		normge0 31146	. . . . . . . . . . . . . . 15 ⊢ (((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ → 0 ≤ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)))
16	14, 15	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → 0 ≤ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)))
17		normcl 31145	. . . . . . . . . . . . . . . 16 ⊢ (((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ∈ ℝ)
18	14, 17	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ∈ ℝ)
19		0re 11264	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
20		letri3 11347	. . . . . . . . . . . . . . . 16 ⊢ (((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ∈ ℝ ∧ 0 ∈ ℝ) → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0 ↔ ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0 ∧ 0 ≤ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)))))
21	20	biimprd 248	. . . . . . . . . . . . . . 15 ⊢ (((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ∈ ℝ ∧ 0 ∈ ℝ) → (((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0 ∧ 0 ≤ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥))) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0))
22	18, 19, 21	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → (((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0 ∧ 0 ≤ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥))) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0))
23	16, 22	sylan2i 606	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → (((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0 ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0))
24	23	anabsi6 670	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0)
25	13, 24	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) ∧ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0)) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0)
26	25	expr 456	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → ((norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0 → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0))
27	1	pjhcli 31438	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐻)‘𝑥) ∈ ℋ)
28		norm-i 31149	. . . . . . . . . . . . 13 ⊢ (((proj_ℎ‘𝐻)‘𝑥) ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0 ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
29	27, 28	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0 ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
30		pjoc2 31459	. . . . . . . . . . . . 13 ⊢ ((𝐻 ∈ C_ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (⊥‘𝐻) ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
31	1, 30	mpan 690	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (𝑥 ∈ (⊥‘𝐻) ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
32	29, 31	bitr4d 282	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0 ↔ 𝑥 ∈ (⊥‘𝐻)))
33	32	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → ((norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0 ↔ 𝑥 ∈ (⊥‘𝐻)))
34		norm-i 31149	. . . . . . . . . . . . 13 ⊢ (((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0 ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
35	14, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0 ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
36		pjoc2 31459	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ C_ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (⊥‘𝐺) ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
37	2, 36	mpan 690	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (𝑥 ∈ (⊥‘𝐺) ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
38	35, 37	bitr4d 282	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0 ↔ 𝑥 ∈ (⊥‘𝐺)))
39	38	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0 ↔ 𝑥 ∈ (⊥‘𝐺)))
40	26, 33, 39	3imtr3d 293	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
41	40	ex 412	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) → (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺))))
42	41	a2i 14	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → (𝑥 ∈ ℋ → (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺))))
43	11, 42	syl5 34	. . . . . 6 ⊢ ((𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → (𝑥 ∈ (⊥‘𝐻) → (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺))))
44	43	pm2.43d 53	. . . . 5 ⊢ ((𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
45	44	alimi 1810	. . . 4 ⊢ (∀𝑥(𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → ∀𝑥(𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
46		df-ral 3061	. . . 4 ⊢ (∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) ↔ ∀𝑥(𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))))
47		df-ss 3967	. . . 4 ⊢ ((⊥‘𝐻) ⊆ (⊥‘𝐺) ↔ ∀𝑥(𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
48	45, 46, 47	3imtr4i 292	. . 3 ⊢ (∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) → (⊥‘𝐻) ⊆ (⊥‘𝐺))
49	2, 1	chsscon3i 31481	. . 3 ⊢ (𝐺 ⊆ 𝐻 ↔ (⊥‘𝐻) ⊆ (⊥‘𝐺))
50	48, 49	sylibr 234	. 2 ⊢ (∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) → 𝐺 ⊆ 𝐻)
51	9, 50	impbii 209	1 ⊢ (𝐺 ⊆ 𝐻 ↔ ∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∩ cin 3949 ⊆ wss 3950 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 0cc0 11156 ≤ cle 11297 ℋchba 30939 ·_ih csp 30942 norm_ℎcno 30943 0_ℎc0v 30944 −_ℎ cmv 30945 C_ℋ cch 30949 ⊥cort 30950 proj_ℎcpjh 30957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cc 10476 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 ax-hilex 31019 ax-hfvadd 31020 ax-hvcom 31021 ax-hvass 31022 ax-hv0cl 31023 ax-hvaddid 31024 ax-hfvmul 31025 ax-hvmulid 31026 ax-hvmulass 31027 ax-hvdistr1 31028 ax-hvdistr2 31029 ax-hvmul0 31030 ax-hfi 31099 ax-his1 31102 ax-his2 31103 ax-his3 31104 ax-his4 31105 ax-hcompl 31222

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-rlim 15526 df-sum 15724 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-cn 23236 df-cnp 23237 df-lm 23238 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cfil 25290 df-cau 25291 df-cmet 25292 df-grpo 30513 df-gid 30514 df-ginv 30515 df-gdiv 30516 df-ablo 30565 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-vs 30619 df-nmcv 30620 df-ims 30621 df-dip 30721 df-ssp 30742 df-ph 30833 df-cbn 30883 df-hnorm 30988 df-hba 30989 df-hvsub 30991 df-hlim 30992 df-hcau 30993 df-sh 31227 df-ch 31241 df-oc 31272 df-ch0 31273 df-shs 31328 df-pjh 31415

This theorem is referenced by: pjssposi 32192

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