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Theorem omlsi 29185
Description: Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
omls.1 𝐴C
omls.2 𝐵S
Assertion
Ref Expression
omlsi ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)

Proof of Theorem omlsi
StepHypRef Expression
1 eqeq1 2826 . 2 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (𝐴 = 𝐵 ↔ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) = 𝐵))
2 eqeq2 2834 . 2 (𝐵 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) → (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) = 𝐵 ↔ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0)))
3 omls.1 . . . 4 𝐴C
4 h0elch 29036 . . . 4 0C
53, 4ifcli 4485 . . 3 if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ∈ C
6 omls.2 . . . 4 𝐵S
7 h0elsh 29037 . . . 4 0S
86, 7ifcli 4485 . . 3 if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∈ S
9 sseq1 3967 . . . . . 6 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (𝐴𝐵 ↔ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 𝐵))
10 fveq2 6652 . . . . . . . 8 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (⊥‘𝐴) = (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0)))
1110ineq2d 4163 . . . . . . 7 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (𝐵 ∩ (⊥‘𝐴)) = (𝐵 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))))
1211eqeq1d 2824 . . . . . 6 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → ((𝐵 ∩ (⊥‘𝐴)) = 0 ↔ (𝐵 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0))
139, 12anbi12d 633 . . . . 5 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) ↔ (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0)))
14 sseq2 3968 . . . . . 6 (𝐵 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) → (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 𝐵 ↔ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0)))
15 ineq1 4155 . . . . . . 7 (𝐵 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) → (𝐵 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))))
1615eqeq1d 2824 . . . . . 6 (𝐵 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) → ((𝐵 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0 ↔ (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0))
1714, 16anbi12d 633 . . . . 5 (𝐵 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) → ((if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 𝐵 ∧ (𝐵 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0) ↔ (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∧ (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0)))
18 sseq1 3967 . . . . . 6 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (0 ⊆ 0 ↔ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 0))
19 fveq2 6652 . . . . . . . 8 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (⊥‘0) = (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0)))
2019ineq2d 4163 . . . . . . 7 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (0 ∩ (⊥‘0)) = (0 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))))
2120eqeq1d 2824 . . . . . 6 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → ((0 ∩ (⊥‘0)) = 0 ↔ (0 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0))
2218, 21anbi12d 633 . . . . 5 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → ((0 ⊆ 0 ∧ (0 ∩ (⊥‘0)) = 0) ↔ (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 0 ∧ (0 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0)))
23 sseq2 3968 . . . . . 6 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) → (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 0 ↔ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0)))
24 ineq1 4155 . . . . . . 7 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) → (0 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))))
2524eqeq1d 2824 . . . . . 6 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) → ((0 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0 ↔ (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0))
2623, 25anbi12d 633 . . . . 5 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) → ((if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 0 ∧ (0 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0) ↔ (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∧ (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0)))
27 ssid 3964 . . . . . 6 0 ⊆ 0
28 ocin 29077 . . . . . . 7 (0S → (0 ∩ (⊥‘0)) = 0)
297, 28ax-mp 5 . . . . . 6 (0 ∩ (⊥‘0)) = 0
3027, 29pm3.2i 474 . . . . 5 (0 ⊆ 0 ∧ (0 ∩ (⊥‘0)) = 0)
3113, 17, 22, 26, 30elimhyp2v 4503 . . . 4 (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∧ (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0)
3231simpli 487 . . 3 if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0)
3331simpri 489 . . 3 (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0
345, 8, 32, 33omlsii 29184 . 2 if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0)
351, 2, 34dedth2v 4499 1 ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  cin 3907  wss 3908  ifcif 4439  cfv 6334   S csh 28709   C cch 28710  cort 28711  0c0h 28716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-cc 9846  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605  ax-mulf 10606  ax-hilex 28780  ax-hfvadd 28781  ax-hvcom 28782  ax-hvass 28783  ax-hv0cl 28784  ax-hvaddid 28785  ax-hfvmul 28786  ax-hvmulid 28787  ax-hvmulass 28788  ax-hvdistr1 28789  ax-hvdistr2 28790  ax-hvmul0 28791  ax-hfi 28860  ax-his1 28863  ax-his2 28864  ax-his3 28865  ax-his4 28866  ax-hcompl 28983
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-acn 9359  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ico 12732  df-icc 12733  df-fz 12886  df-fl 13157  df-seq 13365  df-exp 13426  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-clim 14836  df-rlim 14837  df-rest 16687  df-topgen 16708  df-psmet 20081  df-xmet 20082  df-met 20083  df-bl 20084  df-mopn 20085  df-fbas 20086  df-fg 20087  df-top 21497  df-topon 21514  df-bases 21549  df-cld 21622  df-ntr 21623  df-cls 21624  df-nei 21701  df-lm 21832  df-haus 21918  df-fil 22449  df-fm 22541  df-flim 22542  df-flf 22543  df-cfil 23857  df-cau 23858  df-cmet 23859  df-grpo 28274  df-gid 28275  df-ginv 28276  df-gdiv 28277  df-ablo 28326  df-vc 28340  df-nv 28373  df-va 28376  df-ba 28377  df-sm 28378  df-0v 28379  df-vs 28380  df-nmcv 28381  df-ims 28382  df-ssp 28503  df-ph 28594  df-cbn 28644  df-hnorm 28749  df-hba 28750  df-hvsub 28752  df-hlim 28753  df-hcau 28754  df-sh 28988  df-ch 29002  df-oc 29033  df-ch0 29034
This theorem is referenced by:  pjomli  29216
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