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Mirrors > Home > HSE Home > Th. List > 3oalem6 | Structured version Visualization version GIF version |
Description: Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
3oa.1 | ⊢ 𝐴 ∈ Cℋ |
3oa.2 | ⊢ 𝐵 ∈ Cℋ |
3oa.3 | ⊢ 𝐶 ∈ Cℋ |
3oa.4 | ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) |
3oa.5 | ⊢ 𝑆 = ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴)) |
Ref | Expression |
---|---|
3oalem6 | ⊢ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) ⊆ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3oa.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
2 | 1 | chshii 28931 | . . 3 ⊢ 𝐵 ∈ Sℋ |
3 | 3oa.4 | . . . . . 6 ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) | |
4 | 1 | choccli 29011 | . . . . . . 7 ⊢ (⊥‘𝐵) ∈ Cℋ |
5 | 3oa.1 | . . . . . . . 8 ⊢ 𝐴 ∈ Cℋ | |
6 | 1, 5 | chjcli 29161 | . . . . . . 7 ⊢ (𝐵 ∨ℋ 𝐴) ∈ Cℋ |
7 | 4, 6 | chincli 29164 | . . . . . 6 ⊢ ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) ∈ Cℋ |
8 | 3, 7 | eqeltri 2906 | . . . . 5 ⊢ 𝑅 ∈ Cℋ |
9 | 8 | chshii 28931 | . . . 4 ⊢ 𝑅 ∈ Sℋ |
10 | 3oa.5 | . . . . . . 7 ⊢ 𝑆 = ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴)) | |
11 | 3oa.3 | . . . . . . . . 9 ⊢ 𝐶 ∈ Cℋ | |
12 | 11 | choccli 29011 | . . . . . . . 8 ⊢ (⊥‘𝐶) ∈ Cℋ |
13 | 11, 5 | chjcli 29161 | . . . . . . . 8 ⊢ (𝐶 ∨ℋ 𝐴) ∈ Cℋ |
14 | 12, 13 | chincli 29164 | . . . . . . 7 ⊢ ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴)) ∈ Cℋ |
15 | 10, 14 | eqeltri 2906 | . . . . . 6 ⊢ 𝑆 ∈ Cℋ |
16 | 15 | chshii 28931 | . . . . 5 ⊢ 𝑆 ∈ Sℋ |
17 | 11 | chshii 28931 | . . . . . . 7 ⊢ 𝐶 ∈ Sℋ |
18 | 2, 17 | shscli 29021 | . . . . . 6 ⊢ (𝐵 +ℋ 𝐶) ∈ Sℋ |
19 | 9, 16 | shscli 29021 | . . . . . 6 ⊢ (𝑅 +ℋ 𝑆) ∈ Sℋ |
20 | 18, 19 | shincli 29066 | . . . . 5 ⊢ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)) ∈ Sℋ |
21 | 16, 20 | shscli 29021 | . . . 4 ⊢ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))) ∈ Sℋ |
22 | 9, 21 | shincli 29066 | . . 3 ⊢ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))) ∈ Sℋ |
23 | 2, 22 | shsleji 29074 | . 2 ⊢ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) ⊆ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) |
24 | 16, 20 | shsleji 29074 | . . . . 5 ⊢ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))) ⊆ (𝑆 ∨ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))) |
25 | 1, 11 | chsleji 29162 | . . . . . . . 8 ⊢ (𝐵 +ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶) |
26 | ssrin 4207 | . . . . . . . 8 ⊢ ((𝐵 +ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶) → ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)) ⊆ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))) | |
27 | 25, 26 | ax-mp 5 | . . . . . . 7 ⊢ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)) ⊆ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)) |
28 | 8, 15 | chsleji 29162 | . . . . . . . 8 ⊢ (𝑅 +ℋ 𝑆) ⊆ (𝑅 ∨ℋ 𝑆) |
29 | sslin 4208 | . . . . . . . 8 ⊢ ((𝑅 +ℋ 𝑆) ⊆ (𝑅 ∨ℋ 𝑆) → ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)) ⊆ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))) | |
30 | 28, 29 | ax-mp 5 | . . . . . . 7 ⊢ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)) ⊆ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)) |
31 | 27, 30 | sstri 3973 | . . . . . 6 ⊢ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)) ⊆ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)) |
32 | 1, 11 | chjcli 29161 | . . . . . . . . 9 ⊢ (𝐵 ∨ℋ 𝐶) ∈ Cℋ |
33 | 8, 15 | chjcli 29161 | . . . . . . . . 9 ⊢ (𝑅 ∨ℋ 𝑆) ∈ Cℋ |
34 | 32, 33 | chincli 29164 | . . . . . . . 8 ⊢ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)) ∈ Cℋ |
35 | 34 | chshii 28931 | . . . . . . 7 ⊢ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)) ∈ Sℋ |
36 | 20, 35, 16 | shlej2i 29083 | . . . . . 6 ⊢ (((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)) ⊆ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)) → (𝑆 ∨ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))) ⊆ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)))) |
37 | 31, 36 | ax-mp 5 | . . . . 5 ⊢ (𝑆 ∨ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))) ⊆ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))) |
38 | 24, 37 | sstri 3973 | . . . 4 ⊢ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))) ⊆ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))) |
39 | sslin 4208 | . . . 4 ⊢ ((𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))) ⊆ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))) → (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))) ⊆ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))))) | |
40 | 38, 39 | ax-mp 5 | . . 3 ⊢ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))) ⊆ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)))) |
41 | 15, 34 | chjcli 29161 | . . . . . 6 ⊢ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))) ∈ Cℋ |
42 | 8, 41 | chincli 29164 | . . . . 5 ⊢ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)))) ∈ Cℋ |
43 | 42 | chshii 28931 | . . . 4 ⊢ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)))) ∈ Sℋ |
44 | 22, 43, 2 | shlej2i 29083 | . . 3 ⊢ ((𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))) ⊆ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)))) → (𝐵 ∨ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) ⊆ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆)))))) |
45 | 40, 44 | ax-mp 5 | . 2 ⊢ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) ⊆ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))))) |
46 | 23, 45 | sstri 3973 | 1 ⊢ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) ⊆ (𝐵 ∨ℋ (𝑅 ∩ (𝑆 ∨ℋ ((𝐵 ∨ℋ 𝐶) ∩ (𝑅 ∨ℋ 𝑆))))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Cℋ cch 28633 ⊥cort 28634 +ℋ cph 28635 ∨ℋ chj 28637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 ax-hilex 28703 ax-hfvadd 28704 ax-hvcom 28705 ax-hvass 28706 ax-hv0cl 28707 ax-hvaddid 28708 ax-hfvmul 28709 ax-hvmulid 28710 ax-hvmulass 28711 ax-hvdistr1 28712 ax-hvdistr2 28713 ax-hvmul0 28714 ax-hfi 28783 ax-his1 28786 ax-his2 28787 ax-his3 28788 ax-his4 28789 ax-hcompl 28906 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cn 21763 df-cnp 21764 df-lm 21765 df-haus 21851 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-cau 23786 df-grpo 28197 df-gid 28198 df-ginv 28199 df-gdiv 28200 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-vs 28303 df-nmcv 28304 df-ims 28305 df-dip 28405 df-hnorm 28672 df-hvsub 28675 df-hlim 28676 df-hcau 28677 df-sh 28911 df-ch 28925 df-oc 28956 df-shs 29012 df-chj 29014 |
This theorem is referenced by: 3oai 29372 |
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