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Theorem pjnormssi 30431

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 30428	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ 𝐻 → (((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) = ((proj_ℎ‘(𝐻 ∩ (⊥‘𝐺)))‘𝑥)))
4	1, 2	pjssge0i 30429	. . . . . 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 30430	. . . . 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 238	. . . 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 3106	. 2 ⊢ (𝐺 ⊆ 𝐻 → ∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)))
10	1	choccli 29570	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ C_ℋ
11	10	cheli 29495	. . . . . . 7 ⊢ (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ ℋ)
12		breq2 5074	. . . . . . . . . . . . 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 29681	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ)
15		normge0 29389	. . . . . . . . . . . . . . 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 29388	. . . . . . . . . . . . . . . 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 10908	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
20		letri3 10991	. . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 605	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → (((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0 ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0))
24	23	anabsi6 666	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0)
25	13, 24	sylan2 592	. . . . . . . . . . 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 29681	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐻)‘𝑥) ∈ ℋ)
28		norm-i 29392	. . . . . . . . . . . . 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 29702	. . . . . . . . . . . . 13 ⊢ ((𝐻 ∈ C_ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (⊥‘𝐻) ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
31	1, 30	mpan 686	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (𝑥 ∈ (⊥‘𝐻) ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
32	29, 31	bitr4d 281	. . . . . . . . . . 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 29392	. . . . . . . . . . . . 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 29702	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ C_ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (⊥‘𝐺) ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
37	2, 36	mpan 686	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → (𝑥 ∈ (⊥‘𝐺) ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
38	35, 37	bitr4d 281	. . . . . . . . . . 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 292	. . . . . . . . 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 1815	. . . 4 ⊢ (∀𝑥(𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → ∀𝑥(𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
46		df-ral 3068	. . . 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		dfss2 3903	. . . 4 ⊢ ((⊥‘𝐻) ⊆ (⊥‘𝐺) ↔ ∀𝑥(𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
48	45, 46, 47	3imtr4i 291	. . 3 ⊢ (∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) → (⊥‘𝐻) ⊆ (⊥‘𝐺))
49	2, 1	chsscon3i 29724	. . 3 ⊢ (𝐺 ⊆ 𝐻 ↔ (⊥‘𝐻) ⊆ (⊥‘𝐺))
50	48, 49	sylibr 233	. 2 ⊢ (∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) → 𝐺 ⊆ 𝐻)
51	9, 50	impbii 208	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 205 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∩ cin 3882 ⊆ wss 3883 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 ≤ cle 10941 ℋchba 29182 ·_ih csp 29185 norm_ℎcno 29186 0_ℎc0v 29187 −_ℎ cmv 29188 C_ℋ cch 29192 ⊥cort 29193 proj_ℎcpjh 29200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cc 10122 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882 ax-hilex 29262 ax-hfvadd 29263 ax-hvcom 29264 ax-hvass 29265 ax-hv0cl 29266 ax-hvaddid 29267 ax-hfvmul 29268 ax-hvmulid 29269 ax-hvmulass 29270 ax-hvdistr1 29271 ax-hvdistr2 29272 ax-hvmul0 29273 ax-hfi 29342 ax-his1 29345 ax-his2 29346 ax-his3 29347 ax-his4 29348 ax-hcompl 29465

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-cn 22286 df-cnp 22287 df-lm 22288 df-haus 22374 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cfil 24324 df-cau 24325 df-cmet 24326 df-grpo 28756 df-gid 28757 df-ginv 28758 df-gdiv 28759 df-ablo 28808 df-vc 28822 df-nv 28855 df-va 28858 df-ba 28859 df-sm 28860 df-0v 28861 df-vs 28862 df-nmcv 28863 df-ims 28864 df-dip 28964 df-ssp 28985 df-ph 29076 df-cbn 29126 df-hnorm 29231 df-hba 29232 df-hvsub 29234 df-hlim 29235 df-hcau 29236 df-sh 29470 df-ch 29484 df-oc 29515 df-ch0 29516 df-shs 29571 df-pjh 29658

This theorem is referenced by: pjssposi 30435

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