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Theorem pjnormssi 30221

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 30218	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ 𝐻 → (((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) = ((proj_ℎ‘(𝐻 ∩ (⊥‘𝐺)))‘𝑥)))
4	1, 2	pjssge0i 30219	. . . . . 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 30220	. . . . 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 242	. . . 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 3097	. 2 ⊢ (𝐺 ⊆ 𝐻 → ∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)))
10	1	choccli 29360	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ C_ℋ
11	10	cheli 29285	. . . . . . 7 ⊢ (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ ℋ)
12		breq2 5047	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 29471	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ)
15		normge0 29179	. . . . . . . . . . . . . . 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 29178	. . . . . . . . . . . . . . . 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 10818	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
20		letri3 10901	. . . . . . . . . . . . . . . 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 251	. . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . 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 609	. . . . . . . . . . . . 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 596	. . . . . . . . . . 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 460	. . . . . . . . . 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 29471	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐻)‘𝑥) ∈ ℋ)
28		norm-i 29182	. . . . . . . . . . . . 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 29492	. . . . . . . . . . . . 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 285	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0 ↔ 𝑥 ∈ (⊥‘𝐻)))
33	32	adantr 484	. . . . . . . . . 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 29182	. . . . . . . . . . . . 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 29492	. . . . . . . . . . . . 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 285	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0 ↔ 𝑥 ∈ (⊥‘𝐺)))
39	38	adantr 484	. . . . . . . . . 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 296	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℋ ∧ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
41	40	ex 416	. . . . . . . 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 1819	. . . 4 ⊢ (∀𝑥(𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → ∀𝑥(𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
46		df-ral 3059	. . . 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 3877	. . . 4 ⊢ ((⊥‘𝐻) ⊆ (⊥‘𝐺) ↔ ∀𝑥(𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
48	45, 46, 47	3imtr4i 295	. . 3 ⊢ (∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) → (⊥‘𝐻) ⊆ (⊥‘𝐺))
49	2, 1	chsscon3i 29514	. . 3 ⊢ (𝐺 ⊆ 𝐻 ↔ (⊥‘𝐻) ⊆ (⊥‘𝐺))
50	48, 49	sylibr 237	. 2 ⊢ (∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) → 𝐺 ⊆ 𝐻)
51	9, 50	impbii 212	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 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ∩ cin 3856 ⊆ wss 3857 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 ℝcr 10711 0cc0 10712 ≤ cle 10851 ℋchba 28972 ·_ih csp 28975 norm_ℎcno 28976 0_ℎc0v 28977 −_ℎ cmv 28978 C_ℋ cch 28982 ⊥cort 28983 proj_ℎcpjh 28990

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cc 10032 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 ax-hilex 29052 ax-hfvadd 29053 ax-hvcom 29054 ax-hvass 29055 ax-hv0cl 29056 ax-hvaddid 29057 ax-hfvmul 29058 ax-hvmulid 29059 ax-hvmulass 29060 ax-hvdistr1 29061 ax-hvdistr2 29062 ax-hvmul0 29063 ax-hfi 29132 ax-his1 29135 ax-his2 29136 ax-his3 29137 ax-his4 29138 ax-hcompl 29255

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-omul 8196 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-acn 9541 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-clim 15032 df-rlim 15033 df-sum 15233 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-cn 22096 df-cnp 22097 df-lm 22098 df-haus 22184 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cfil 24124 df-cau 24125 df-cmet 24126 df-grpo 28546 df-gid 28547 df-ginv 28548 df-gdiv 28549 df-ablo 28598 df-vc 28612 df-nv 28645 df-va 28648 df-ba 28649 df-sm 28650 df-0v 28651 df-vs 28652 df-nmcv 28653 df-ims 28654 df-dip 28754 df-ssp 28775 df-ph 28866 df-cbn 28916 df-hnorm 29021 df-hba 29022 df-hvsub 29024 df-hlim 29025 df-hcau 29026 df-sh 29260 df-ch 29274 df-oc 29305 df-ch0 29306 df-shs 29361 df-pjh 29448

This theorem is referenced by: pjssposi 30225

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