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Mirrors > Home > HSE Home > Th. List > chscllem1 | Structured version Visualization version GIF version |
Description: Lemma for chscl 27884. (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 2610 | . . . 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 | ffvelrnda 6267 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) ∈ (𝐴 +ℋ 𝐵)) |
6 | chscl.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Cℋ ) | |
7 | chsh 27465 | . . . . . . . . . 10 ⊢ (𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ ) | |
8 | 6, 7 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Sℋ ) |
9 | chsh 27465 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
10 | 2, 9 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Sℋ ) |
11 | shocsh 27527 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ ) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (⊥‘𝐴) ∈ Sℋ ) |
13 | chscl.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) | |
14 | shless 27602 | . . . . . . . . 9 ⊢ (((𝐵 ∈ Sℋ ∧ (⊥‘𝐴) ∈ Sℋ ∧ 𝐴 ∈ Sℋ ) ∧ 𝐵 ⊆ (⊥‘𝐴)) → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) | |
15 | 8, 12, 10, 13, 14 | syl31anc 1321 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) |
16 | shscom 27562 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) | |
17 | 10, 8, 16 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) |
18 | shscom 27562 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ (⊥‘𝐴) ∈ Sℋ ) → (𝐴 +ℋ (⊥‘𝐴)) = ((⊥‘𝐴) +ℋ 𝐴)) | |
19 | 10, 12, 18 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℋ (⊥‘𝐴)) = ((⊥‘𝐴) +ℋ 𝐴)) |
20 | 15, 17, 19 | 3sstr4d 3611 | . . . . . . 7 ⊢ (𝜑 → (𝐴 +ℋ 𝐵) ⊆ (𝐴 +ℋ (⊥‘𝐴))) |
21 | 20 | sselda 3568 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐻‘𝑛) ∈ (𝐴 +ℋ 𝐵)) → (𝐻‘𝑛) ∈ (𝐴 +ℋ (⊥‘𝐴))) |
22 | 5, 21 | syldan 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) ∈ (𝐴 +ℋ (⊥‘𝐴))) |
23 | pjpreeq 27641 | . . . . 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 691 | . . . 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 222 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴 ∧ ∃𝑥 ∈ (⊥‘𝐴)(𝐻‘𝑛) = (((projℎ‘𝐴)‘(𝐻‘𝑛)) +ℎ 𝑥))) |
26 | 25 | simpld 474 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴) |
27 | chscl.6 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛))) | |
28 | 26, 27 | fmptd 6292 | 1 ⊢ (𝜑 → 𝐹:ℕ⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℕcn 10897 +ℎ cva 27161 ⇝𝑣 chli 27168 Sℋ csh 27169 Cℋ cch 27170 ⊥cort 27171 +ℋ cph 27172 projℎcpjh 27178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his2 27324 ax-his3 27325 ax-his4 27326 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-grpo 26731 df-ablo 26783 df-hvsub 27212 df-sh 27448 df-ch 27462 df-oc 27493 df-ch0 27494 df-shs 27551 df-pjh 27638 |
This theorem is referenced by: chscllem2 27881 chscllem3 27882 chscllem4 27883 |
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