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Mirrors > Home > HSE Home > Th. List > 3oalem5 | 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 |
---|---|
3oalem5 | ⊢ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) = ((𝐵 ∨ℋ 𝑅) ∩ (𝐶 ∨ℋ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3oa.4 | . . . . 5 ⊢ 𝑅 = ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) | |
2 | 1 | 3oalem4 31495 | . . . 4 ⊢ 𝑅 ⊆ (⊥‘𝐵) |
3 | 3oa.2 | . . . . . . . 8 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | choccli 31137 | . . . . . . 7 ⊢ (⊥‘𝐵) ∈ Cℋ |
5 | 3oa.1 | . . . . . . . 8 ⊢ 𝐴 ∈ Cℋ | |
6 | 3, 5 | chjcli 31287 | . . . . . . 7 ⊢ (𝐵 ∨ℋ 𝐴) ∈ Cℋ |
7 | 4, 6 | chincli 31290 | . . . . . 6 ⊢ ((⊥‘𝐵) ∩ (𝐵 ∨ℋ 𝐴)) ∈ Cℋ |
8 | 1, 7 | eqeltri 2825 | . . . . 5 ⊢ 𝑅 ∈ Cℋ |
9 | 8, 3 | osumi 31472 | . . . 4 ⊢ (𝑅 ⊆ (⊥‘𝐵) → (𝑅 +ℋ 𝐵) = (𝑅 ∨ℋ 𝐵)) |
10 | 2, 9 | ax-mp 5 | . . 3 ⊢ (𝑅 +ℋ 𝐵) = (𝑅 ∨ℋ 𝐵) |
11 | 3 | chshii 31057 | . . . 4 ⊢ 𝐵 ∈ Sℋ |
12 | 8 | chshii 31057 | . . . 4 ⊢ 𝑅 ∈ Sℋ |
13 | 11, 12 | shscomi 31193 | . . 3 ⊢ (𝐵 +ℋ 𝑅) = (𝑅 +ℋ 𝐵) |
14 | 3, 8 | chjcomi 31298 | . . 3 ⊢ (𝐵 ∨ℋ 𝑅) = (𝑅 ∨ℋ 𝐵) |
15 | 10, 13, 14 | 3eqtr4i 2766 | . 2 ⊢ (𝐵 +ℋ 𝑅) = (𝐵 ∨ℋ 𝑅) |
16 | 3oa.5 | . . . . 5 ⊢ 𝑆 = ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴)) | |
17 | 16 | 3oalem4 31495 | . . . 4 ⊢ 𝑆 ⊆ (⊥‘𝐶) |
18 | 3oa.3 | . . . . . . . 8 ⊢ 𝐶 ∈ Cℋ | |
19 | 18 | choccli 31137 | . . . . . . 7 ⊢ (⊥‘𝐶) ∈ Cℋ |
20 | 18, 5 | chjcli 31287 | . . . . . . 7 ⊢ (𝐶 ∨ℋ 𝐴) ∈ Cℋ |
21 | 19, 20 | chincli 31290 | . . . . . 6 ⊢ ((⊥‘𝐶) ∩ (𝐶 ∨ℋ 𝐴)) ∈ Cℋ |
22 | 16, 21 | eqeltri 2825 | . . . . 5 ⊢ 𝑆 ∈ Cℋ |
23 | 22, 18 | osumi 31472 | . . . 4 ⊢ (𝑆 ⊆ (⊥‘𝐶) → (𝑆 +ℋ 𝐶) = (𝑆 ∨ℋ 𝐶)) |
24 | 17, 23 | ax-mp 5 | . . 3 ⊢ (𝑆 +ℋ 𝐶) = (𝑆 ∨ℋ 𝐶) |
25 | 18 | chshii 31057 | . . . 4 ⊢ 𝐶 ∈ Sℋ |
26 | 22 | chshii 31057 | . . . 4 ⊢ 𝑆 ∈ Sℋ |
27 | 25, 26 | shscomi 31193 | . . 3 ⊢ (𝐶 +ℋ 𝑆) = (𝑆 +ℋ 𝐶) |
28 | 18, 22 | chjcomi 31298 | . . 3 ⊢ (𝐶 ∨ℋ 𝑆) = (𝑆 ∨ℋ 𝐶) |
29 | 24, 27, 28 | 3eqtr4i 2766 | . 2 ⊢ (𝐶 +ℋ 𝑆) = (𝐶 ∨ℋ 𝑆) |
30 | 15, 29 | ineq12i 4212 | 1 ⊢ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) = ((𝐵 ∨ℋ 𝑅) ∩ (𝐶 ∨ℋ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∩ cin 3948 ⊆ wss 3949 ‘cfv 6553 (class class class)co 7426 Cℋ cch 30759 ⊥cort 30760 +ℋ cph 30761 ∨ℋ chj 30763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cc 10466 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226 ax-hilex 30829 ax-hfvadd 30830 ax-hvcom 30831 ax-hvass 30832 ax-hv0cl 30833 ax-hvaddid 30834 ax-hfvmul 30835 ax-hvmulid 30836 ax-hvmulass 30837 ax-hvdistr1 30838 ax-hvdistr2 30839 ax-hvmul0 30840 ax-hfi 30909 ax-his1 30912 ax-his2 30913 ax-his3 30914 ax-his4 30915 ax-hcompl 31032 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-omul 8498 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-acn 9973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 df-sum 15673 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-cn 23151 df-cnp 23152 df-lm 23153 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cfil 25203 df-cau 25204 df-cmet 25205 df-grpo 30323 df-gid 30324 df-ginv 30325 df-gdiv 30326 df-ablo 30375 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-vs 30429 df-nmcv 30430 df-ims 30431 df-dip 30531 df-ssp 30552 df-ph 30643 df-cbn 30693 df-hnorm 30798 df-hba 30799 df-hvsub 30801 df-hlim 30802 df-hcau 30803 df-sh 31037 df-ch 31051 df-oc 31082 df-ch0 31083 df-shs 31138 df-chj 31140 df-pjh 31225 |
This theorem is referenced by: 3oai 31498 |
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