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Mirrors > Home > HSE Home > Th. List > mdsymlem8 | Structured version Visualization version GIF version |
Description: Lemma for mdsymi 31351. Lemma 4(ii) of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mdsymlem1.1 | ⊢ 𝐴 ∈ Cℋ |
mdsymlem1.2 | ⊢ 𝐵 ∈ Cℋ |
mdsymlem1.3 | ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝) |
Ref | Expression |
---|---|
mdsymlem8 | ⊢ ((𝐴 ≠ 0ℋ ∧ 𝐵 ≠ 0ℋ) → (𝐵 𝑀ℋ* 𝐴 ↔ 𝐴 𝑀ℋ* 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdsymlem1.1 | . . . . . . 7 ⊢ 𝐴 ∈ Cℋ | |
2 | mdsymlem1.2 | . . . . . . 7 ⊢ 𝐵 ∈ Cℋ | |
3 | 1, 2 | chjcomi 30408 | . . . . . 6 ⊢ (𝐴 ∨ℋ 𝐵) = (𝐵 ∨ℋ 𝐴) |
4 | 3 | sseq2i 3973 | . . . . 5 ⊢ (𝑝 ⊆ (𝐴 ∨ℋ 𝐵) ↔ 𝑝 ⊆ (𝐵 ∨ℋ 𝐴)) |
5 | atelch 31284 | . . . . . . . . . 10 ⊢ (𝑞 ∈ HAtoms → 𝑞 ∈ Cℋ ) | |
6 | atelch 31284 | . . . . . . . . . 10 ⊢ (𝑟 ∈ HAtoms → 𝑟 ∈ Cℋ ) | |
7 | chjcom 30446 | . . . . . . . . . 10 ⊢ ((𝑞 ∈ Cℋ ∧ 𝑟 ∈ Cℋ ) → (𝑞 ∨ℋ 𝑟) = (𝑟 ∨ℋ 𝑞)) | |
8 | 5, 6, 7 | syl2an 596 | . . . . . . . . 9 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → (𝑞 ∨ℋ 𝑟) = (𝑟 ∨ℋ 𝑞)) |
9 | 8 | sseq2d 3976 | . . . . . . . 8 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ↔ 𝑝 ⊆ (𝑟 ∨ℋ 𝑞))) |
10 | ancom 461 | . . . . . . . . 9 ⊢ ((𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵) ↔ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴)) | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → ((𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵) ↔ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴))) |
12 | 9, 11 | anbi12d 631 | . . . . . . 7 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → ((𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵)) ↔ (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴)))) |
13 | 12 | 2rexbiia 3209 | . . . . . 6 ⊢ (∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵)) ↔ ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴))) |
14 | rexcom 3273 | . . . . . 6 ⊢ (∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴)) ↔ ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴))) | |
15 | 13, 14 | bitri 274 | . . . . 5 ⊢ (∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵)) ↔ ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴))) |
16 | 4, 15 | imbi12i 350 | . . . 4 ⊢ ((𝑝 ⊆ (𝐴 ∨ℋ 𝐵) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵))) ↔ (𝑝 ⊆ (𝐵 ∨ℋ 𝐴) → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴)))) |
17 | 16 | ralbii 3096 | . . 3 ⊢ (∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐴 ∨ℋ 𝐵) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵))) ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐵 ∨ℋ 𝐴) → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴)))) |
18 | 17 | a1i 11 | . 2 ⊢ ((𝐴 ≠ 0ℋ ∧ 𝐵 ≠ 0ℋ) → (∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐴 ∨ℋ 𝐵) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵))) ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐵 ∨ℋ 𝐴) → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴))))) |
19 | mdsymlem1.3 | . . 3 ⊢ 𝐶 = (𝐴 ∨ℋ 𝑝) | |
20 | 1, 2, 19 | mdsymlem7 31349 | . 2 ⊢ ((𝐴 ≠ 0ℋ ∧ 𝐵 ≠ 0ℋ) → (𝐵 𝑀ℋ* 𝐴 ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐴 ∨ℋ 𝐵) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵))))) |
21 | eqid 2736 | . . . 4 ⊢ (𝐵 ∨ℋ 𝑝) = (𝐵 ∨ℋ 𝑝) | |
22 | 2, 1, 21 | mdsymlem7 31349 | . . 3 ⊢ ((𝐵 ≠ 0ℋ ∧ 𝐴 ≠ 0ℋ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐵 ∨ℋ 𝐴) → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴))))) |
23 | 22 | ancoms 459 | . 2 ⊢ ((𝐴 ≠ 0ℋ ∧ 𝐵 ≠ 0ℋ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐵 ∨ℋ 𝐴) → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴))))) |
24 | 18, 20, 23 | 3bitr4d 310 | 1 ⊢ ((𝐴 ≠ 0ℋ ∧ 𝐵 ≠ 0ℋ) → (𝐵 𝑀ℋ* 𝐴 ↔ 𝐴 𝑀ℋ* 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 ⊆ wss 3910 class class class wbr 5105 (class class class)co 7356 Cℋ cch 29869 ∨ℋ chj 29873 0ℋc0h 29875 HAtomscat 29905 𝑀ℋ* cdmd 29907 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cc 10370 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 ax-addf 11129 ax-mulf 11130 ax-hilex 29939 ax-hfvadd 29940 ax-hvcom 29941 ax-hvass 29942 ax-hv0cl 29943 ax-hvaddid 29944 ax-hfvmul 29945 ax-hvmulid 29946 ax-hvmulass 29947 ax-hvdistr1 29948 ax-hvdistr2 29949 ax-hvmul0 29950 ax-hfi 30019 ax-his1 30022 ax-his2 30023 ax-his3 30024 ax-his4 30025 ax-hcompl 30142 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8647 df-map 8766 df-pm 8767 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-fi 9346 df-sup 9377 df-inf 9378 df-oi 9445 df-card 9874 df-acn 9877 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13267 df-ico 13269 df-icc 13270 df-fz 13424 df-fzo 13567 df-fl 13696 df-seq 13906 df-exp 13967 df-hash 14230 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-clim 15369 df-rlim 15370 df-sum 15570 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-starv 17147 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-hom 17156 df-cco 17157 df-rest 17303 df-topn 17304 df-0g 17322 df-gsum 17323 df-topgen 17324 df-pt 17325 df-prds 17328 df-xrs 17383 df-qtop 17388 df-imas 17389 df-xps 17391 df-mre 17465 df-mrc 17466 df-acs 17468 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-submnd 18601 df-mulg 18871 df-cntz 19095 df-cmn 19562 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-fbas 20791 df-fg 20792 df-cnfld 20795 df-top 22241 df-topon 22258 df-topsp 22280 df-bases 22294 df-cld 22368 df-ntr 22369 df-cls 22370 df-nei 22447 df-cn 22576 df-cnp 22577 df-lm 22578 df-haus 22664 df-tx 22911 df-hmeo 23104 df-fil 23195 df-fm 23287 df-flim 23288 df-flf 23289 df-xms 23671 df-ms 23672 df-tms 23673 df-cfil 24617 df-cau 24618 df-cmet 24619 df-grpo 29433 df-gid 29434 df-ginv 29435 df-gdiv 29436 df-ablo 29485 df-vc 29499 df-nv 29532 df-va 29535 df-ba 29536 df-sm 29537 df-0v 29538 df-vs 29539 df-nmcv 29540 df-ims 29541 df-dip 29641 df-ssp 29662 df-ph 29753 df-cbn 29803 df-hnorm 29908 df-hba 29909 df-hvsub 29911 df-hlim 29912 df-hcau 29913 df-sh 30147 df-ch 30161 df-oc 30192 df-ch0 30193 df-shs 30248 df-span 30249 df-chj 30250 df-chsup 30251 df-pjh 30335 df-cv 31219 df-dmd 31221 df-at 31278 |
This theorem is referenced by: mdsymi 31351 |
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