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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 30984 | . . . . . 6 ⊢ (⊥‘𝐴) ∈ Cℋ |
4 | golem1.2 | . . . . . . 7 ⊢ 𝐵 ∈ Cℋ | |
5 | 2, 4 | chincli 31137 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
6 | 3, 5 | chjcli 31134 | . . . . 5 ⊢ ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐵)) ∈ Cℋ |
7 | 1, 6 | eqeltri 2821 | . . . 4 ⊢ 𝐹 ∈ Cℋ |
8 | golem1.5 | . . . . 5 ⊢ 𝐺 = ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐶)) | |
9 | 4 | choccli 30984 | . . . . . 6 ⊢ (⊥‘𝐵) ∈ Cℋ |
10 | golem1.3 | . . . . . . 7 ⊢ 𝐶 ∈ Cℋ | |
11 | 4, 10 | chincli 31137 | . . . . . 6 ⊢ (𝐵 ∩ 𝐶) ∈ Cℋ |
12 | 9, 11 | chjcli 31134 | . . . . 5 ⊢ ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐶)) ∈ Cℋ |
13 | 8, 12 | eqeltri 2821 | . . . 4 ⊢ 𝐺 ∈ Cℋ |
14 | golem1.6 | . . . . 5 ⊢ 𝐻 = ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐴)) | |
15 | 10 | choccli 30984 | . . . . . 6 ⊢ (⊥‘𝐶) ∈ Cℋ |
16 | 10, 2 | chincli 31137 | . . . . . 6 ⊢ (𝐶 ∩ 𝐴) ∈ Cℋ |
17 | 15, 16 | chjcli 31134 | . . . . 5 ⊢ ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐴)) ∈ Cℋ |
18 | 14, 17 | eqeltri 2821 | . . . 4 ⊢ 𝐻 ∈ Cℋ |
19 | 7, 13, 18 | stm1add3i 31924 | . . 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 31948 | . . . 4 ⊢ (𝑓 ∈ States → (((𝑓‘𝐹) + (𝑓‘𝐺)) + (𝑓‘𝐻)) = (((𝑓‘𝐷) + (𝑓‘𝑅)) + (𝑓‘𝑆))) |
24 | 23 | eqeq1d 2726 | . . 3 ⊢ (𝑓 ∈ States → ((((𝑓‘𝐹) + (𝑓‘𝐺)) + (𝑓‘𝐻)) = 3 ↔ (((𝑓‘𝐷) + (𝑓‘𝑅)) + (𝑓‘𝑆)) = 3)) |
25 | 19, 24 | sylibd 238 | . 2 ⊢ (𝑓 ∈ States → ((𝑓‘((𝐹 ∩ 𝐺) ∩ 𝐻)) = 1 → (((𝑓‘𝐷) + (𝑓‘𝑅)) + (𝑓‘𝑆)) = 3)) |
26 | 4, 2 | chincli 31137 | . . . . 5 ⊢ (𝐵 ∩ 𝐴) ∈ Cℋ |
27 | 9, 26 | chjcli 31134 | . . . 4 ⊢ ((⊥‘𝐵) ∨ℋ (𝐵 ∩ 𝐴)) ∈ Cℋ |
28 | 20, 27 | eqeltri 2821 | . . 3 ⊢ 𝐷 ∈ Cℋ |
29 | 10, 4 | chincli 31137 | . . . . 5 ⊢ (𝐶 ∩ 𝐵) ∈ Cℋ |
30 | 15, 29 | chjcli 31134 | . . . 4 ⊢ ((⊥‘𝐶) ∨ℋ (𝐶 ∩ 𝐵)) ∈ Cℋ |
31 | 21, 30 | eqeltri 2821 | . . 3 ⊢ 𝑅 ∈ Cℋ |
32 | 2, 10 | chincli 31137 | . . . . 5 ⊢ (𝐴 ∩ 𝐶) ∈ Cℋ |
33 | 3, 32 | chjcli 31134 | . . . 4 ⊢ ((⊥‘𝐴) ∨ℋ (𝐴 ∩ 𝐶)) ∈ Cℋ |
34 | 22, 33 | eqeltri 2821 | . . 3 ⊢ 𝑆 ∈ Cℋ |
35 | 28, 31, 34 | stadd3i 31925 | . 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 1533 ∈ wcel 2098 ∩ cin 3939 ‘cfv 6533 (class class class)co 7401 1c1 11106 + caddc 11108 3c3 12264 Cℋ cch 30606 ⊥cort 30607 ∨ℋ chj 30610 Statescst 30639 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cc 10425 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185 ax-hilex 30676 ax-hfvadd 30677 ax-hvcom 30678 ax-hvass 30679 ax-hv0cl 30680 ax-hvaddid 30681 ax-hfvmul 30682 ax-hvmulid 30683 ax-hvmulass 30684 ax-hvdistr1 30685 ax-hvdistr2 30686 ax-hvmul0 30687 ax-hfi 30756 ax-his1 30759 ax-his2 30760 ax-his3 30761 ax-his4 30762 ax-hcompl 30879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-acn 9932 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-submnd 18701 df-mulg 18983 df-cntz 19218 df-cmn 19687 df-psmet 21215 df-xmet 21216 df-met 21217 df-bl 21218 df-mopn 21219 df-fbas 21220 df-fg 21221 df-cnfld 21224 df-top 22706 df-topon 22723 df-topsp 22745 df-bases 22759 df-cld 22833 df-ntr 22834 df-cls 22835 df-nei 22912 df-cn 23041 df-cnp 23042 df-lm 23043 df-haus 23129 df-tx 23376 df-hmeo 23569 df-fil 23660 df-fm 23752 df-flim 23753 df-flf 23754 df-xms 24136 df-ms 24137 df-tms 24138 df-cfil 25093 df-cau 25094 df-cmet 25095 df-grpo 30170 df-gid 30171 df-ginv 30172 df-gdiv 30173 df-ablo 30222 df-vc 30236 df-nv 30269 df-va 30272 df-ba 30273 df-sm 30274 df-0v 30275 df-vs 30276 df-nmcv 30277 df-ims 30278 df-dip 30378 df-ssp 30399 df-ph 30490 df-cbn 30540 df-hnorm 30645 df-hba 30646 df-hvsub 30648 df-hlim 30649 df-hcau 30650 df-sh 30884 df-ch 30898 df-oc 30929 df-ch0 30930 df-chj 30987 df-st 31888 |
This theorem is referenced by: goeqi 31950 |
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