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Mirrors > Home > HSE Home > Th. List > chscllem1 | Structured version Visualization version GIF version |
Description: Lemma for chscl 29418. (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 2821 | . . . 4 ⊢ ((projℎ‘𝐴)‘(𝐻‘𝑛)) = ((projℎ‘𝐴)‘(𝐻‘𝑛)) | |
2 | chscl.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Cℋ ) | |
3 | 2 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ Cℋ ) |
4 | chscl.4 | . . . . . . 7 ⊢ (𝜑 → 𝐻:ℕ⟶(𝐴 +ℋ 𝐵)) | |
5 | 4 | ffvelrnda 6851 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) ∈ (𝐴 +ℋ 𝐵)) |
6 | chscl.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ Cℋ ) | |
7 | chsh 29001 | . . . . . . . . . 10 ⊢ (𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ ) | |
8 | 6, 7 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Sℋ ) |
9 | chsh 29001 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
10 | 2, 9 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Sℋ ) |
11 | shocsh 29061 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ ) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (⊥‘𝐴) ∈ Sℋ ) |
13 | chscl.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) | |
14 | shless 29136 | . . . . . . . . 9 ⊢ (((𝐵 ∈ Sℋ ∧ (⊥‘𝐴) ∈ Sℋ ∧ 𝐴 ∈ Sℋ ) ∧ 𝐵 ⊆ (⊥‘𝐴)) → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) | |
15 | 8, 12, 10, 13, 14 | syl31anc 1369 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) |
16 | shscom 29096 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) | |
17 | 10, 8, 16 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℋ 𝐵) = (𝐵 +ℋ 𝐴)) |
18 | shscom 29096 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Sℋ ∧ (⊥‘𝐴) ∈ Sℋ ) → (𝐴 +ℋ (⊥‘𝐴)) = ((⊥‘𝐴) +ℋ 𝐴)) | |
19 | 10, 12, 18 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 +ℋ (⊥‘𝐴)) = ((⊥‘𝐴) +ℋ 𝐴)) |
20 | 15, 17, 19 | 3sstr4d 4014 | . . . . . . 7 ⊢ (𝜑 → (𝐴 +ℋ 𝐵) ⊆ (𝐴 +ℋ (⊥‘𝐴))) |
21 | 20 | sselda 3967 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐻‘𝑛) ∈ (𝐴 +ℋ 𝐵)) → (𝐻‘𝑛) ∈ (𝐴 +ℋ (⊥‘𝐴))) |
22 | 5, 21 | syldan 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) ∈ (𝐴 +ℋ (⊥‘𝐴))) |
23 | pjpreeq 29175 | . . . . 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 586 | . . . 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 235 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴 ∧ ∃𝑥 ∈ (⊥‘𝐴)(𝐻‘𝑛) = (((projℎ‘𝐴)‘(𝐻‘𝑛)) +ℎ 𝑥))) |
26 | 25 | simpld 497 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((projℎ‘𝐴)‘(𝐻‘𝑛)) ∈ 𝐴) |
27 | chscl.6 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((projℎ‘𝐴)‘(𝐻‘𝑛))) | |
28 | 26, 27 | fmptd 6878 | 1 ⊢ (𝜑 → 𝐹:ℕ⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℕcn 11638 +ℎ cva 28697 ⇝𝑣 chli 28704 Sℋ csh 28705 Cℋ cch 28706 ⊥cort 28707 +ℋ cph 28708 projℎcpjh 28714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvmulass 28784 ax-hvdistr1 28785 ax-hvdistr2 28786 ax-hvmul0 28787 ax-hfi 28856 ax-his2 28860 ax-his3 28861 ax-his4 28862 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-grpo 28270 df-ablo 28322 df-hvsub 28748 df-sh 28984 df-ch 28998 df-oc 29029 df-ch0 29030 df-shs 29085 df-pjh 29172 |
This theorem is referenced by: chscllem2 29415 chscllem3 29416 chscllem4 29417 |
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