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Mirrors > Home > HSE Home > Th. List > chscllem1 | Structured version Visualization version GIF version |
Description: Lemma for chscl 28500. (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 2622 | . . . 4 ⊢ ((projℎ‘𝐴)‘(𝐻‘𝑛)) = ((projℎ‘𝐴)‘(𝐻‘𝑛)) | |
2 | chscl.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Cℋ ) | |
3 | 2 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ Cℋ ) |
4 | chscl.4 | . . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) | |
5 | 4 | ffvelrnda 6359 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) ∈ (𝐴 +ℋ 𝐵)) |
6 | chscl.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Cℋ ) | |
7 | chsh 28081 | . . . . . . . . . 10 ⊢ (𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ ) | |
8 | 6, 7 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Sℋ ) |
9 | chsh 28081 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
10 | 2, 9 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Sℋ ) |
11 | shocsh 28143 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ ) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (⊥‘𝐴) ∈ Sℋ ) |
13 | chscl.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) | |
14 | shless 28218 | . . . . . . . . 9 ⊢ (((𝐵 ∈ Sℋ ∧ (⊥‘𝐴) ∈ Sℋ ∧ 𝐴 ∈ Sℋ ) ∧ 𝐵 ⊆ (⊥‘𝐴)) → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) | |
15 | 8, 12, 10, 13, 14 | syl31anc 1329 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) |
16 | shscom 28178 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) | |
17 | 10, 8, 16 | syl2anc 693 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) |
18 | shscom 28178 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ (⊥‘𝐴) ∈ Sℋ ) → (𝐴 +ℋ (⊥‘𝐴)) = ((⊥‘𝐴) +ℋ 𝐴)) | |
19 | 10, 12, 18 | syl2anc 693 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℋ (⊥‘𝐴)) = ((⊥‘𝐴) +ℋ 𝐴)) |
20 | 15, 17, 19 | 3sstr4d 3648 | . . . . . . 7 ⊢ (𝜑 → (𝐴 +ℋ 𝐵) ⊆ (𝐴 +ℋ (⊥‘𝐴))) |
21 | 20 | sselda 3603 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐻‘𝑛) ∈ (𝐴 +ℋ 𝐵)) → (𝐻‘𝑛) ∈ (𝐴 +ℋ (⊥‘𝐴))) |
22 | 5, 21 | syldan 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) ∈ (𝐴 +ℋ (⊥‘𝐴))) |
23 | pjpreeq 28257 | . . . . 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 693 | . . . 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 223 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴 ∧ ∃𝑥 ∈ (⊥‘𝐴)(𝐻‘𝑛) = (((projℎ‘𝐴)‘(𝐻‘𝑛)) +ℎ 𝑥))) |
26 | 25 | simpld 475 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴) |
27 | chscl.6 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛))) | |
28 | 26, 27 | fmptd 6385 | 1 ⊢ (𝜑 → 𝐹:ℕ⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℕcn 11020 +ℎ cva 27777 ⇝𝑣 chli 27784 Sℋ csh 27785 Cℋ cch 27786 ⊥cort 27787 +ℋ cph 27788 projℎcpjh 27794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his2 27940 ax-his3 27941 ax-his4 27942 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-grpo 27347 df-ablo 27399 df-hvsub 27828 df-sh 28064 df-ch 28078 df-oc 28109 df-ch0 28110 df-shs 28167 df-pjh 28254 |
This theorem is referenced by: chscllem2 28497 chscllem3 28498 chscllem4 28499 |
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