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Theorem pjnormssi 32262

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 32259	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ 𝐻 → (((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) = ((proj_ℎ‘(𝐻 ∩ (⊥‘𝐺)))‘𝑥)))
4	1, 2	pjssge0i 32260	. . . . . 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 32261	. . . . 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 3129	. 2 ⊢ (𝐺 ⊆ 𝐻 → ∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)))
10	1	choccli 31401	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ C_ℋ
11	10	cheli 31326	. . . . . . 7 ⊢ (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ ℋ)
12		breq2 5104	. . . . . . . . . . . . 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 31512	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ)
15		normge0 31220	. . . . . . . . . . . . . . 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 31219	. . . . . . . . . . . . . . . 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 11148	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
20		letri3 11232	. . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 607	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → (((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0 ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0))
24	23	anabsi6 671	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0)
25	13, 24	sylan2 594	. . . . . . . . . . 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 31512	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐻)‘𝑥) ∈ ℋ)
28		norm-i 31223	. . . . . . . . . . . . 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 31533	. . . . . . . . . . . . 13 ⊢ ((𝐻 ∈ C_ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (⊥‘𝐻) ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
31	1, 30	mpan 691	. . . . . . . . . . . 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 31223	. . . . . . . . . . . . 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 31533	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ C_ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (⊥‘𝐺) ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
37	2, 36	mpan 691	. . . . . . . . . . . 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 1813	. . . 4 ⊢ (∀𝑥(𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → ∀𝑥(𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
46		df-ral 3053	. . . 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 3920	. . . 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 31555	. . 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 1540 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∩ cin 3902 ⊆ wss 3903 class class class wbr 5100 ‘cfv 6502 (class class class)co 7370 ℝcr 11039 0cc0 11040 ≤ cle 11181 ℋchba 31013 ·_ih csp 31016 norm_ℎcno 31017 0_ℎc0v 31018 −_ℎ cmv 31019 C_ℋ cch 31023 ⊥cort 31024 proj_ℎcpjh 31031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-inf2 9564 ax-cc 10359 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118 ax-addf 11119 ax-mulf 11120 ax-hilex 31093 ax-hfvadd 31094 ax-hvcom 31095 ax-hvass 31096 ax-hv0cl 31097 ax-hvaddid 31098 ax-hfvmul 31099 ax-hvmulid 31100 ax-hvmulass 31101 ax-hvdistr1 31102 ax-hvdistr2 31103 ax-hvmul0 31104 ax-hfi 31173 ax-his1 31176 ax-his2 31177 ax-his3 31178 ax-his4 31179 ax-hcompl 31296

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-oadd 8413 df-omul 8414 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-fi 9328 df-sup 9359 df-inf 9360 df-oi 9429 df-card 9865 df-acn 9868 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-q 12876 df-rp 12920 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13279 df-ico 13281 df-icc 13282 df-fz 13438 df-fzo 13585 df-fl 13726 df-seq 13939 df-exp 13999 df-hash 14268 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-clim 15425 df-rlim 15426 df-sum 15624 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-starv 17206 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-unif 17214 df-hom 17215 df-cco 17216 df-rest 17356 df-topn 17357 df-0g 17375 df-gsum 17376 df-topgen 17377 df-pt 17378 df-prds 17381 df-xrs 17437 df-qtop 17442 df-imas 17443 df-xps 17445 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-submnd 18723 df-mulg 19015 df-cntz 19263 df-cmn 19728 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-fbas 21323 df-fg 21324 df-cnfld 21327 df-top 22855 df-topon 22872 df-topsp 22894 df-bases 22907 df-cld 22980 df-ntr 22981 df-cls 22982 df-nei 23059 df-cn 23188 df-cnp 23189 df-lm 23190 df-haus 23276 df-tx 23523 df-hmeo 23716 df-fil 23807 df-fm 23899 df-flim 23900 df-flf 23901 df-xms 24281 df-ms 24282 df-tms 24283 df-cfil 25228 df-cau 25229 df-cmet 25230 df-grpo 30587 df-gid 30588 df-ginv 30589 df-gdiv 30590 df-ablo 30639 df-vc 30653 df-nv 30686 df-va 30689 df-ba 30690 df-sm 30691 df-0v 30692 df-vs 30693 df-nmcv 30694 df-ims 30695 df-dip 30795 df-ssp 30816 df-ph 30907 df-cbn 30957 df-hnorm 31062 df-hba 31063 df-hvsub 31065 df-hlim 31066 df-hcau 31067 df-sh 31301 df-ch 31315 df-oc 31346 df-ch0 31347 df-shs 31402 df-pjh 31489

This theorem is referenced by: pjssposi 32266

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