![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > golem2 | Structured version Visualization version GIF version |
Description: Lemma for Godowski's equation. (Contributed by NM, 13-Nov-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
golem1.1 | ⊢ 𝐴 ∈ Cℋ |
golem1.2 | ⊢ 𝐵 ∈ Cℋ |
golem1.3 | ⊢ 𝐶 ∈ Cℋ |
golem1.4 | ⊢ 𝐹 = ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵)) |
golem1.5 | ⊢ 𝐺 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐶)) |
golem1.6 | ⊢ 𝐻 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐴)) |
golem1.7 | ⊢ 𝐷 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐴)) |
golem1.8 | ⊢ 𝑅 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐵)) |
golem1.9 | ⊢ 𝑆 = ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐶)) |
Ref | Expression |
---|---|
golem2 | ⊢ (𝑓 ∈ States → ((𝑓‘((𝐹 ∩ 𝐺) ∩ 𝐻)) = 1 → (𝑓‘𝐷) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | golem1.4 | . . . . 5 ⊢ 𝐹 = ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵)) | |
2 | golem1.1 | . . . . . . 7 ⊢ 𝐴 ∈ Cℋ | |
3 | 2 | choccli 28475 | . . . . . 6 ⊢ (⊥‘𝐴) ∈ Cℋ |
4 | golem1.2 | . . . . . . 7 ⊢ 𝐵 ∈ Cℋ | |
5 | 2, 4 | chincli 28628 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
6 | 3, 5 | chjcli 28625 | . . . . 5 ⊢ ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵)) ∈ Cℋ |
7 | 1, 6 | eqeltri 2835 | . . . 4 ⊢ 𝐹 ∈ Cℋ |
8 | golem1.5 | . . . . 5 ⊢ 𝐺 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐶)) | |
9 | 4 | choccli 28475 | . . . . . 6 ⊢ (⊥‘𝐵) ∈ Cℋ |
10 | golem1.3 | . . . . . . 7 ⊢ 𝐶 ∈ Cℋ | |
11 | 4, 10 | chincli 28628 | . . . . . 6 ⊢ (𝐵 ∩ 𝐶) ∈ Cℋ |
12 | 9, 11 | chjcli 28625 | . . . . 5 ⊢ ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐶)) ∈ Cℋ |
13 | 8, 12 | eqeltri 2835 | . . . 4 ⊢ 𝐺 ∈ Cℋ |
14 | golem1.6 | . . . . 5 ⊢ 𝐻 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐴)) | |
15 | 10 | choccli 28475 | . . . . . 6 ⊢ (⊥‘𝐶) ∈ Cℋ |
16 | 10, 2 | chincli 28628 | . . . . . 6 ⊢ (𝐶 ∩ 𝐴) ∈ Cℋ |
17 | 15, 16 | chjcli 28625 | . . . . 5 ⊢ ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐴)) ∈ Cℋ |
18 | 14, 17 | eqeltri 2835 | . . . 4 ⊢ 𝐻 ∈ Cℋ |
19 | 7, 13, 18 | stm1add3i 29415 | . . 3 ⊢ (𝑓 ∈ States → ((𝑓‘((𝐹 ∩ 𝐺) ∩ 𝐻)) = 1 → (((𝑓‘𝐹) + (𝑓‘𝐺)) + (𝑓‘𝐻)) = 3)) |
20 | golem1.7 | . . . . 5 ⊢ 𝐷 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐴)) | |
21 | golem1.8 | . . . . 5 ⊢ 𝑅 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐵)) | |
22 | golem1.9 | . . . . 5 ⊢ 𝑆 = ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐶)) | |
23 | 2, 4, 10, 1, 8, 14, 20, 21, 22 | golem1 29439 | . . . 4 ⊢ (𝑓 ∈ States → (((𝑓‘𝐹) + (𝑓‘𝐺)) + (𝑓‘𝐻)) = (((𝑓‘𝐷) + (𝑓‘𝑅)) + (𝑓‘𝑆))) |
24 | 23 | eqeq1d 2762 | . . 3 ⊢ (𝑓 ∈ States → ((((𝑓‘𝐹) + (𝑓‘𝐺)) + (𝑓‘𝐻)) = 3 ↔ (((𝑓‘𝐷) + (𝑓‘𝑅)) + (𝑓‘𝑆)) = 3)) |
25 | 19, 24 | sylibd 229 | . 2 ⊢ (𝑓 ∈ States → ((𝑓‘((𝐹 ∩ 𝐺) ∩ 𝐻)) = 1 → (((𝑓‘𝐷) + (𝑓‘𝑅)) + (𝑓‘𝑆)) = 3)) |
26 | 4, 2 | chincli 28628 | . . . . 5 ⊢ (𝐵 ∩ 𝐴) ∈ Cℋ |
27 | 9, 26 | chjcli 28625 | . . . 4 ⊢ ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐴)) ∈ Cℋ |
28 | 20, 27 | eqeltri 2835 | . . 3 ⊢ 𝐷 ∈ Cℋ |
29 | 10, 4 | chincli 28628 | . . . . 5 ⊢ (𝐶 ∩ 𝐵) ∈ Cℋ |
30 | 15, 29 | chjcli 28625 | . . . 4 ⊢ ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐵)) ∈ Cℋ |
31 | 21, 30 | eqeltri 2835 | . . 3 ⊢ 𝑅 ∈ Cℋ |
32 | 2, 10 | chincli 28628 | . . . . 5 ⊢ (𝐴 ∩ 𝐶) ∈ Cℋ |
33 | 3, 32 | chjcli 28625 | . . . 4 ⊢ ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐶)) ∈ Cℋ |
34 | 22, 33 | eqeltri 2835 | . . 3 ⊢ 𝑆 ∈ Cℋ |
35 | 28, 31, 34 | stadd3i 29416 | . 2 ⊢ (𝑓 ∈ States → ((((𝑓‘𝐷) + (𝑓‘𝑅)) + (𝑓‘𝑆)) = 3 → (𝑓‘𝐷) = 1)) |
36 | 25, 35 | syld 47 | 1 ⊢ (𝑓 ∈ States → ((𝑓‘((𝐹 ∩ 𝐺) ∩ 𝐻)) = 1 → (𝑓‘𝐷) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 ∩ cin 3714 ‘cfv 6049 (class class class)co 6813 1c1 10129 + caddc 10131 3c3 11263 Cℋ cch 28095 ⊥cort 28096 ∨ℋ chj 28099 Statescst 28128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cc 9449 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208 ax-hilex 28165 ax-hfvadd 28166 ax-hvcom 28167 ax-hvass 28168 ax-hv0cl 28169 ax-hvaddid 28170 ax-hfvmul 28171 ax-hvmulid 28172 ax-hvmulass 28173 ax-hvdistr1 28174 ax-hvdistr2 28175 ax-hvmul0 28176 ax-hfi 28245 ax-his1 28248 ax-his2 28249 ax-his3 28250 ax-his4 28251 ax-hcompl 28368 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-omul 7734 df-er 7911 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-fi 8482 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-acn 8958 df-cda 9182 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ioo 12372 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-rlim 14419 df-sum 14616 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-hom 16168 df-cco 16169 df-rest 16285 df-topn 16286 df-0g 16304 df-gsum 16305 df-topgen 16306 df-pt 16307 df-prds 16310 df-xrs 16364 df-qtop 16369 df-imas 16370 df-xps 16372 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-mulg 17742 df-cntz 17950 df-cmn 18395 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-fbas 19945 df-fg 19946 df-cnfld 19949 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cld 21025 df-ntr 21026 df-cls 21027 df-nei 21104 df-cn 21233 df-cnp 21234 df-lm 21235 df-haus 21321 df-tx 21567 df-hmeo 21760 df-fil 21851 df-fm 21943 df-flim 21944 df-flf 21945 df-xms 22326 df-ms 22327 df-tms 22328 df-cfil 23253 df-cau 23254 df-cmet 23255 df-grpo 27656 df-gid 27657 df-ginv 27658 df-gdiv 27659 df-ablo 27708 df-vc 27723 df-nv 27756 df-va 27759 df-ba 27760 df-sm 27761 df-0v 27762 df-vs 27763 df-nmcv 27764 df-ims 27765 df-dip 27865 df-ssp 27886 df-ph 27977 df-cbn 28028 df-hnorm 28134 df-hba 28135 df-hvsub 28137 df-hlim 28138 df-hcau 28139 df-sh 28373 df-ch 28387 df-oc 28418 df-ch0 28419 df-chj 28478 df-st 29379 |
This theorem is referenced by: goeqi 29441 |
Copyright terms: Public domain | W3C validator |