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| Description: Lemma for chscl 31660. (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 2737 | . . . 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 31243 | . . . . . . . . . 10 ⊢ (𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ ) | |
| 8 | 6, 7 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Sℋ ) | 
| 9 | chsh 31243 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
| 10 | 2, 9 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Sℋ ) | 
| 11 | shocsh 31303 | . . . . . . . . . 10 ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ ) | |
| 12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (⊥‘𝐴) ∈ Sℋ ) | 
| 13 | chscl.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ (⊥‘𝐴)) | |
| 14 | shless 31378 | . . . . . . . . 9 ⊢ (((𝐵 ∈ Sℋ ∧ (⊥‘𝐴) ∈ Sℋ ∧ 𝐴 ∈ Sℋ ) ∧ 𝐵 ⊆ (⊥‘𝐴)) → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) | |
| 15 | 8, 12, 10, 13, 14 | syl31anc 1375 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 +ℋ 𝐴) ⊆ ((⊥‘𝐴) +ℋ 𝐴)) | 
| 16 | shscom 31338 | . . . . . . . . 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 31338 | . . . . . . . . 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 4039 | . . . . . . 7 ⊢ (𝜑 → (𝐴 +ℋ 𝐵) ⊆ (𝐴 +ℋ (⊥‘𝐴))) | 
| 21 | 20 | sselda 3983 | . . . . . 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 31417 | . . . . 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 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℕcn 12266 +ℎ cva 30939 ⇝𝑣 chli 30946 Sℋ csh 30947 Cℋ cch 30948 ⊥cort 30949 +ℋ cph 30950 projℎcpjh 30956 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvmulass 31026 ax-hvdistr1 31027 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his2 31102 ax-his3 31103 ax-his4 31104 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-grpo 30512 df-ablo 30564 df-hvsub 30990 df-sh 31226 df-ch 31240 df-oc 31271 df-ch0 31272 df-shs 31327 df-pjh 31414 | 
| This theorem is referenced by: chscllem2 31657 chscllem3 31658 chscllem4 31659 | 
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