![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > chscllem1 | Structured version Visualization version GIF version |
Description: Lemma for chscl 31670. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chscl.1 | ⊢ (𝜑 → 𝐴 ∈ Cℋ ) |
chscl.2 | ⊢ (𝜑 → 𝐵 ∈ Cℋ ) |
chscl.3 | ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) |
chscl.4 | ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) |
chscl.5 | ⊢ (𝜑 → 𝐻 ⇝𝑣 𝑢) |
chscl.6 | ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛))) |
Ref | Expression |
---|---|
chscllem1 | ⊢ (𝜑 → 𝐹:ℕ⟶𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . 4 ⊢ ((projℎ‘𝐴)‘(𝐻‘𝑛)) = ((projℎ‘𝐴)‘(𝐻‘𝑛)) | |
2 | chscl.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Cℋ ) | |
3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ Cℋ ) |
4 | chscl.4 | . . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) | |
5 | 4 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) ∈ (𝐴 +ℋ 𝐵)) |
6 | chscl.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Cℋ ) | |
7 | chsh 31253 | . . . . . . . . . 10 ⊢ (𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ ) | |
8 | 6, 7 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Sℋ ) |
9 | chsh 31253 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
10 | 2, 9 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Sℋ ) |
11 | shocsh 31313 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ ) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (⊥‘𝐴) ∈ Sℋ ) |
13 | chscl.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) | |
14 | shless 31388 | . . . . . . . . 9 ⊢ (((𝐵 ∈ Sℋ ∧ (⊥‘𝐴) ∈ Sℋ ∧ 𝐴 ∈ Sℋ ) ∧ 𝐵 ⊆ (⊥‘𝐴)) → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) | |
15 | 8, 12, 10, 13, 14 | syl31anc 1372 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) |
16 | shscom 31348 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) | |
17 | 10, 8, 16 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) |
18 | shscom 31348 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ (⊥‘𝐴) ∈ Sℋ ) → (𝐴 +ℋ (⊥‘𝐴)) = ((⊥‘𝐴) +ℋ 𝐴)) | |
19 | 10, 12, 18 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℋ (⊥‘𝐴)) = ((⊥‘𝐴) +ℋ 𝐴)) |
20 | 15, 17, 19 | 3sstr4d 4043 | . . . . . . 7 ⊢ (𝜑 → (𝐴 +ℋ 𝐵) ⊆ (𝐴 +ℋ (⊥‘𝐴))) |
21 | 20 | sselda 3995 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐻‘𝑛) ∈ (𝐴 +ℋ 𝐵)) → (𝐻‘𝑛) ∈ (𝐴 +ℋ (⊥‘𝐴))) |
22 | 5, 21 | syldan 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) ∈ (𝐴 +ℋ (⊥‘𝐴))) |
23 | pjpreeq 31427 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ (𝐻‘𝑛) ∈ (𝐴 +ℋ (⊥‘𝐴))) → (((projℎ‘𝐴)‘(𝐻‘𝑛)) = ((projℎ‘𝐴)‘(𝐻‘𝑛)) ↔ (((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴 ∧ ∃𝑥 ∈ (⊥‘𝐴)(𝐻‘𝑛) = (((projℎ‘𝐴)‘(𝐻‘𝑛)) +ℎ 𝑥)))) | |
24 | 3, 22, 23 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((projℎ‘𝐴)‘(𝐻‘𝑛)) = ((projℎ‘𝐴)‘(𝐻‘𝑛)) ↔ (((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴 ∧ ∃𝑥 ∈ (⊥‘𝐴)(𝐻‘𝑛) = (((projℎ‘𝐴)‘(𝐻‘𝑛)) +ℎ 𝑥)))) |
25 | 1, 24 | mpbii 233 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴 ∧ ∃𝑥 ∈ (⊥‘𝐴)(𝐻‘𝑛) = (((projℎ‘𝐴)‘(𝐻‘𝑛)) +ℎ 𝑥))) |
26 | 25 | simpld 494 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴) |
27 | chscl.6 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛))) | |
28 | 26, 27 | fmptd 7134 | 1 ⊢ (𝜑 → 𝐹:ℕ⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℕcn 12264 +ℎ cva 30949 ⇝𝑣 chli 30956 Sℋ csh 30957 Cℋ cch 30958 ⊥cort 30959 +ℋ cph 30960 projℎcpjh 30966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-hilex 31028 ax-hfvadd 31029 ax-hvcom 31030 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvmulass 31036 ax-hvdistr1 31037 ax-hvdistr2 31038 ax-hvmul0 31039 ax-hfi 31108 ax-his2 31112 ax-his3 31113 ax-his4 31114 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-grpo 30522 df-ablo 30574 df-hvsub 31000 df-sh 31236 df-ch 31250 df-oc 31281 df-ch0 31282 df-shs 31337 df-pjh 31424 |
This theorem is referenced by: chscllem2 31667 chscllem3 31668 chscllem4 31669 |
Copyright terms: Public domain | W3C validator |