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Theorem pjnormssi 32681

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 32678	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ 𝐻 → (((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) = ((proj_ℎ‘(𝐻 ∩ (⊥‘𝐺)))‘𝑥)))
4	1, 2	pjssge0i 32679	. . . . . 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 48	. . . . 5 ⊢ (𝑥 ∈ ℋ → (𝐺 ⊆ 𝐻 → 0 ≤ ((((proj_ℎ‘𝐻)‘𝑥) −_ℎ ((proj_ℎ‘𝐺)‘𝑥)) ·_ih 𝑥)))
6	1, 2	pjdifnormi 32680	. . . . 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 33	. . 3 ⊢ (𝐺 ⊆ 𝐻 → (𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))))
9	8	ralrimiv 3153	. 2 ⊢ (𝐺 ⊆ 𝐻 → ∀𝑥 ∈ ℋ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)))
10	1	choccli 31820	. . . . . . . 8 ⊢ (⊥‘𝐻) ∈ C_ℋ
11	10	cheli 31745	. . . . . . 7 ⊢ (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ ℋ)
12		breq2 5107	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 31931	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ)
15		normge0 31639	. . . . . . . . . . . . . . 15 ⊢ (((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ → 0 ≤ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)))
16	14, 15	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℋ → 0 ≤ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)))
17		normcl 31638	. . . . . . . . . . . . . . . 16 ⊢ (((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ∈ ℝ)
18	14, 17	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ∈ ℝ)
19		0re 11259	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
20		letri3 11344	. . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . 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 618	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → (((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0 ∧ 𝑥 ∈ ℋ) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0))
24	23	anabsi6 683	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ 0) → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0)
25	13, 24	sylan2 605	. . . . . . . . . . 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 462	. . . . . . . . . 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 31931	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘𝐻)‘𝑥) ∈ ℋ)
28		norm-i 31642	. . . . . . . . . . . . 13 ⊢ (((proj_ℎ‘𝐻)‘𝑥) ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0 ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
29	27, 28	syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) = 0 ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
30		pjoc2 31952	. . . . . . . . . . . . 13 ⊢ ((𝐻 ∈ C_ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (⊥‘𝐻) ↔ ((proj_ℎ‘𝐻)‘𝑥) = 0_ℎ))
31	1, 30	mpan 703	. . . . . . . . . . . 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 486	. . . . . . . . . 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 31642	. . . . . . . . . . . . 13 ⊢ (((proj_ℎ‘𝐺)‘𝑥) ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0 ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
35	14, 34	syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) = 0 ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
36		pjoc2 31952	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ C_ℋ ∧ 𝑥 ∈ ℋ) → (𝑥 ∈ (⊥‘𝐺) ↔ ((proj_ℎ‘𝐺)‘𝑥) = 0_ℎ))
37	2, 36	mpan 703	. . . . . . . . . . . 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 486	. . . . . . . . . 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 418	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → ((norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥)) → (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺))))
42	41	a2i 15	. . . . . . 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 35	. . . . . 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 54	. . . . 5 ⊢ ((𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → (𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
45	44	alimi 1844	. . . 4 ⊢ (∀𝑥(𝑥 ∈ ℋ → (norm_ℎ‘((proj_ℎ‘𝐺)‘𝑥)) ≤ (norm_ℎ‘((proj_ℎ‘𝐻)‘𝑥))) → ∀𝑥(𝑥 ∈ (⊥‘𝐻) → 𝑥 ∈ (⊥‘𝐺)))
46		df-ral 3077	. . . 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 3916	. . . 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 31974	. . 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∀wal 1568 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∩ cin 3898 ⊆ wss 3899 class class class wbr 5103 ‘cfv 6535 (class class class)co 7416 ℝcr 11148 0cc0 11149 ≤ cle 11293 ℋchba 31432 ·_ih csp 31435 norm_ℎcno 31436 0_ℎc0v 31437 −_ℎ cmv 31438 C_ℋ cch 31442 ⊥cort 31443 proj_ℎcpjh 31450

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cc 10462 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 ax-mulf 11229 ax-hilex 31512 ax-hfvadd 31513 ax-hvcom 31514 ax-hvass 31515 ax-hv0cl 31516 ax-hvaddid 31517 ax-hfvmul 31518 ax-hvmulid 31519 ax-hvmulass 31520 ax-hvdistr1 31521 ax-hvdistr2 31522 ax-hvmul0 31523 ax-hfi 31592 ax-his1 31595 ax-his2 31596 ax-his3 31597 ax-his4 31598 ax-hcompl 31715

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9489 df-card 9969 df-acn 9972 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-q 13023 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-ioo 13427 df-ico 13429 df-icc 13430 df-fz 13587 df-fzo 13735 df-fl 13878 df-seq 14091 df-exp 14151 df-hash 14420 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-clim 15600 df-rlim 15601 df-sum 15799 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-rest 17532 df-topn 17533 df-0g 17551 df-gsum 17552 df-topgen 17553 df-pt 17554 df-prds 17557 df-xrs 17613 df-qtop 17618 df-imas 17619 df-xps 17621 df-mre 17695 df-mrc 17696 df-acs 17698 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-submnd 18918 df-mulg 19217 df-cntz 19470 df-cmn 19935 df-psmet 21609 df-xmet 21610 df-met 21611 df-bl 21612 df-mopn 21613 df-fbas 21614 df-fg 21615 df-cnfld 21618 df-top 23151 df-topon 23168 df-topsp 23190 df-bases 23203 df-cld 23276 df-ntr 23277 df-cls 23278 df-nei 23355 df-cn 23484 df-cnp 23485 df-lm 23486 df-haus 23572 df-tx 23820 df-hmeo 24013 df-fil 24104 df-fm 24196 df-flim 24197 df-flf 24198 df-xms 24578 df-ms 24579 df-tms 24580 df-cfil 25515 df-cau 25516 df-cmet 25517 df-grpo 31006 df-gid 31007 df-ginv 31008 df-gdiv 31009 df-ablo 31058 df-vc 31072 df-nv 31105 df-va 31108 df-ba 31109 df-sm 31110 df-0v 31111 df-vs 31112 df-nmcv 31113 df-ims 31114 df-dip 31214 df-ssp 31235 df-ph 31326 df-cbn 31376 df-hnorm 31481 df-hba 31482 df-hvsub 31484 df-hlim 31485 df-hcau 31486 df-sh 31720 df-ch 31734 df-oc 31765 df-ch0 31766 df-shs 31821 df-pjh 31908

This theorem is used by: pjssposi 32685

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