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Theorem omlsi 28818
 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 2829 . 2 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (𝐴 = 𝐵 ↔ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) = 𝐵))
2 eqeq2 2836 . 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 28667 . . . 4 0C
53, 4ifcli 4352 . . 3 if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ∈ C
6 omls.2 . . . 4 𝐵S
7 h0elsh 28668 . . . 4 0S
86, 7ifcli 4352 . . 3 if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∈ S
9 sseq1 3851 . . . . . 6 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (𝐴𝐵 ↔ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 𝐵))
10 fveq2 6433 . . . . . . . 8 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (⊥‘𝐴) = (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0)))
1110ineq2d 4041 . . . . . . 7 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (𝐵 ∩ (⊥‘𝐴)) = (𝐵 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))))
1211eqeq1d 2827 . . . . . 6 (𝐴 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → ((𝐵 ∩ (⊥‘𝐴)) = 0 ↔ (𝐵 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0))
139, 12anbi12d 626 . . . . 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 3852 . . . . . 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 4034 . . . . . . 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 2827 . . . . . 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 626 . . . . 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 3851 . . . . . 6 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (0 ⊆ 0 ↔ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ 0))
19 fveq2 6433 . . . . . . . 8 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (⊥‘0) = (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0)))
2019ineq2d 4041 . . . . . . 7 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → (0 ∩ (⊥‘0)) = (0 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))))
2120eqeq1d 2827 . . . . . 6 (0 = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) → ((0 ∩ (⊥‘0)) = 0 ↔ (0 ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0))
2218, 21anbi12d 626 . . . . 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 3852 . . . . . 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 4034 . . . . . . 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 2827 . . . . . 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 626 . . . . 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 3848 . . . . . 6 0 ⊆ 0
28 ocin 28710 . . . . . . 7 (0S → (0 ∩ (⊥‘0)) = 0)
297, 28ax-mp 5 . . . . . 6 (0 ∩ (⊥‘0)) = 0
3027, 29pm3.2i 464 . . . . 5 (0 ⊆ 0 ∧ (0 ∩ (⊥‘0)) = 0)
3113, 17, 22, 26, 30elimhyp2v 4370 . . . 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 478 . . 3 if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) ⊆ if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0)
3331simpri 481 . . 3 (if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0) ∩ (⊥‘if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0))) = 0
345, 8, 32, 33omlsii 28817 . 2 if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐴, 0) = if((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0), 𝐵, 0)
351, 2, 34dedth2v 4366 1 ((𝐴𝐵 ∧ (𝐵 ∩ (⊥‘𝐴)) = 0) → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ∩ cin 3797   ⊆ wss 3798  ifcif 4306  ‘cfv 6123   Sℋ csh 28340   Cℋ cch 28341  ⊥cort 28342  0ℋc0h 28347 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cc 9572  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330  ax-addf 10331  ax-mulf 10332  ax-hilex 28411  ax-hfvadd 28412  ax-hvcom 28413  ax-hvass 28414  ax-hv0cl 28415  ax-hvaddid 28416  ax-hfvmul 28417  ax-hvmulid 28418  ax-hvmulass 28419  ax-hvdistr1 28420  ax-hvdistr2 28421  ax-hvmul0 28422  ax-hfi 28491  ax-his1 28494  ax-his2 28495  ax-his3 28496  ax-his4 28497  ax-hcompl 28614 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-omul 7831  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fi 8586  df-sup 8617  df-inf 8618  df-oi 8684  df-card 9078  df-acn 9081  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-n0 11619  df-z 11705  df-uz 11969  df-q 12072  df-rp 12113  df-xneg 12232  df-xadd 12233  df-xmul 12234  df-ico 12469  df-icc 12470  df-fz 12620  df-fl 12888  df-seq 13096  df-exp 13155  df-cj 14216  df-re 14217  df-im 14218  df-sqrt 14352  df-abs 14353  df-clim 14596  df-rlim 14597  df-rest 16436  df-topgen 16457  df-psmet 20098  df-xmet 20099  df-met 20100  df-bl 20101  df-mopn 20102  df-fbas 20103  df-fg 20104  df-top 21069  df-topon 21086  df-bases 21121  df-cld 21194  df-ntr 21195  df-cls 21196  df-nei 21273  df-lm 21404  df-haus 21490  df-fil 22020  df-fm 22112  df-flim 22113  df-flf 22114  df-cfil 23423  df-cau 23424  df-cmet 23425  df-grpo 27903  df-gid 27904  df-ginv 27905  df-gdiv 27906  df-ablo 27955  df-vc 27969  df-nv 28002  df-va 28005  df-ba 28006  df-sm 28007  df-0v 28008  df-vs 28009  df-nmcv 28010  df-ims 28011  df-ssp 28132  df-ph 28223  df-cbn 28274  df-hnorm 28380  df-hba 28381  df-hvsub 28383  df-hlim 28384  df-hcau 28385  df-sh 28619  df-ch 28633  df-oc 28664  df-ch0 28665 This theorem is referenced by:  pjomli  28849
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