mdsymlem8 - Hilbert Space Explorer

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Theorem mdsymlem8 32699

Description: Lemma for mdsymi 32700. Lemma 4(ii) of [Maeda] p. 168. (Contributed by NM, 3-Jul-2004.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
mdsymlem1.1	⊢ 𝐴 ∈ C_ℋ
mdsymlem1.2	⊢ 𝐵 ∈ C_ℋ
mdsymlem1.3	⊢ 𝐶 = (𝐴 ∨_ℋ 𝑝)

**Assertion**
Ref	Expression
mdsymlem8	⊢ ((𝐴 ≠ 0_ℋ ∧ 𝐵 ≠ 0_ℋ) → (𝐵 𝑀_ℋ^* 𝐴 ↔ 𝐴 𝑀_ℋ^* 𝐵))

Distinct variable groups: 𝐴,𝑝 𝐵,𝑝 Allowed substitution hint: 𝐶(𝑝)

Proof of Theorem mdsymlem8 Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdsymlem1.1	. . . . . . 7 ⊢ 𝐴 ∈ C_ℋ
2		mdsymlem1.2	. . . . . . 7 ⊢ 𝐵 ∈ C_ℋ
3	1, 2	chjcomi 31757	. . . . . 6 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
4	3	sseq2i 3974	. . . . 5 ⊢ (𝑝 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ 𝑝 ⊆ (𝐵 ∨_ℋ 𝐴))
5		atelch 32633	. . . . . . . . . 10 ⊢ (𝑞 ∈ HAtoms → 𝑞 ∈ C_ℋ )
6		atelch 32633	. . . . . . . . . 10 ⊢ (𝑟 ∈ HAtoms → 𝑟 ∈ C_ℋ )
7		chjcom 31795	. . . . . . . . . 10 ⊢ ((𝑞 ∈ C_ℋ ∧ 𝑟 ∈ C_ℋ ) → (𝑞 ∨_ℋ 𝑟) = (𝑟 ∨_ℋ 𝑞))
8	5, 6, 7	syl2an 607	. . . . . . . . 9 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → (𝑞 ∨_ℋ 𝑟) = (𝑟 ∨_ℋ 𝑞))
9	8	sseq2d 3977	. . . . . . . 8 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → (𝑝 ⊆ (𝑞 ∨_ℋ 𝑟) ↔ 𝑝 ⊆ (𝑟 ∨_ℋ 𝑞)))
10		ancom 465	. . . . . . . . 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 643	. . . . . . 7 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → ((𝑝 ⊆ (𝑞 ∨_ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵)) ↔ (𝑝 ⊆ (𝑟 ∨_ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴))))
13	12	2rexbiia 3232	. . . . . 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 3300	. . . . . 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 278	. . . . 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 353	. . . 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 3117	. . 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 ≠ 0_ℋ ∧ 𝐵 ≠ 0_ℋ) → (∀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 32698	. 2 ⊢ ((𝐴 ≠ 0_ℋ ∧ 𝐵 ≠ 0_ℋ) → (𝐵 𝑀_ℋ^* 𝐴 ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐴 ∨_ℋ 𝐵) → ∃𝑞 ∈ HAtoms ∃𝑟 ∈ HAtoms (𝑝 ⊆ (𝑞 ∨_ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵)))))
21		eqid 2769	. . . 4 ⊢ (𝐵 ∨_ℋ 𝑝) = (𝐵 ∨_ℋ 𝑝)
22	2, 1, 21	mdsymlem7 32698	. . 3 ⊢ ((𝐵 ≠ 0_ℋ ∧ 𝐴 ≠ 0_ℋ) → (𝐴 𝑀_ℋ^* 𝐵 ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐵 ∨_ℋ 𝐴) → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨_ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴)))))
23	22	ancoms 463	. 2 ⊢ ((𝐴 ≠ 0_ℋ ∧ 𝐵 ≠ 0_ℋ) → (𝐴 𝑀_ℋ^* 𝐵 ↔ ∀𝑝 ∈ HAtoms (𝑝 ⊆ (𝐵 ∨_ℋ 𝐴) → ∃𝑟 ∈ HAtoms ∃𝑞 ∈ HAtoms (𝑝 ⊆ (𝑟 ∨_ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴)))))
24	18, 20, 23	3bitr4d 314	1 ⊢ ((𝐴 ≠ 0_ℋ ∧ 𝐵 ≠ 0_ℋ) → (𝐵 𝑀_ℋ^* 𝐴 ↔ 𝐴 𝑀_ℋ^* 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ⊆ wss 3913 class class class wbr 5110 (class class class)co 7408 C_ℋ cch 31218 ∨_ℋ chj 31222 0_ℋc0h 31224 HAtomscat 31254 𝑀_ℋ^* cdmd 31256

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-inf2 9606 ax-cc 10415 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174 ax-addf 11175 ax-mulf 11176 ax-hilex 31288 ax-hfvadd 31289 ax-hvcom 31290 ax-hvass 31291 ax-hv0cl 31292 ax-hvaddid 31293 ax-hfvmul 31294 ax-hvmulid 31295 ax-hvmulass 31296 ax-hvdistr1 31297 ax-hvdistr2 31298 ax-hvmul0 31299 ax-hfi 31368 ax-his1 31371 ax-his2 31372 ax-his3 31373 ax-his4 31374 ax-hcompl 31491

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-omul 8454 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-acn 9924 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13372 df-ico 13374 df-icc 13375 df-fz 13532 df-fzo 13679 df-fl 13821 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-rlim 15536 df-sum 15734 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17471 df-topn 17472 df-0g 17490 df-gsum 17491 df-topgen 17492 df-pt 17493 df-prds 17496 df-xrs 17552 df-qtop 17557 df-imas 17558 df-xps 17560 df-mre 17634 df-mrc 17635 df-acs 17637 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-mulg 19130 df-cntz 19383 df-cmn 19848 df-psmet 21479 df-xmet 21480 df-met 21481 df-bl 21482 df-mopn 21483 df-fbas 21484 df-fg 21485 df-cnfld 21488 df-top 23016 df-topon 23033 df-topsp 23055 df-bases 23068 df-cld 23141 df-ntr 23142 df-cls 23143 df-nei 23220 df-cn 23349 df-cnp 23350 df-lm 23351 df-haus 23437 df-tx 23684 df-hmeo 23877 df-fil 23968 df-fm 24060 df-flim 24061 df-flf 24062 df-xms 24442 df-ms 24443 df-tms 24444 df-cfil 25379 df-cau 25380 df-cmet 25381 df-grpo 30782 df-gid 30783 df-ginv 30784 df-gdiv 30785 df-ablo 30834 df-vc 30848 df-nv 30881 df-va 30884 df-ba 30885 df-sm 30886 df-0v 30887 df-vs 30888 df-nmcv 30889 df-ims 30890 df-dip 30990 df-ssp 31011 df-ph 31102 df-cbn 31152 df-hnorm 31257 df-hba 31258 df-hvsub 31260 df-hlim 31261 df-hcau 31262 df-sh 31496 df-ch 31510 df-oc 31541 df-ch0 31542 df-shs 31597 df-span 31598 df-chj 31599 df-chsup 31600 df-pjh 31684 df-cv 32568 df-dmd 32570 df-at 32627

This theorem is referenced by: mdsymi 32700

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