mdsymlem8 - Hilbert Space Explorer

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Theorem mdsymlem8 31927

Description: Lemma for mdsymi 31928. 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 30985	. . . . . 6 ⊢ (𝐴 ∨_ℋ 𝐵) = (𝐵 ∨_ℋ 𝐴)
4	3	sseq2i 4012	. . . . 5 ⊢ (𝑝 ⊆ (𝐴 ∨_ℋ 𝐵) ↔ 𝑝 ⊆ (𝐵 ∨_ℋ 𝐴))
5		atelch 31861	. . . . . . . . . 10 ⊢ (𝑞 ∈ HAtoms → 𝑞 ∈ C_ℋ )
6		atelch 31861	. . . . . . . . . 10 ⊢ (𝑟 ∈ HAtoms → 𝑟 ∈ C_ℋ )
7		chjcom 31023	. . . . . . . . . 10 ⊢ ((𝑞 ∈ C_ℋ ∧ 𝑟 ∈ C_ℋ ) → (𝑞 ∨_ℋ 𝑟) = (𝑟 ∨_ℋ 𝑞))
8	5, 6, 7	syl2an 595	. . . . . . . . 9 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → (𝑞 ∨_ℋ 𝑟) = (𝑟 ∨_ℋ 𝑞))
9	8	sseq2d 4015	. . . . . . . 8 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → (𝑝 ⊆ (𝑞 ∨_ℋ 𝑟) ↔ 𝑝 ⊆ (𝑟 ∨_ℋ 𝑞)))
10		ancom 460	. . . . . . . . 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 630	. . . . . . 7 ⊢ ((𝑞 ∈ HAtoms ∧ 𝑟 ∈ HAtoms) → ((𝑝 ⊆ (𝑞 ∨_ℋ 𝑟) ∧ (𝑞 ⊆ 𝐴 ∧ 𝑟 ⊆ 𝐵)) ↔ (𝑝 ⊆ (𝑟 ∨_ℋ 𝑞) ∧ (𝑟 ⊆ 𝐵 ∧ 𝑞 ⊆ 𝐴))))
13	12	2rexbiia 3214	. . . . . 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 3286	. . . . . 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 349	. . . 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 3092	. . 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 31926	. 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 2731	. . . 4 ⊢ (𝐵 ∨_ℋ 𝑝) = (𝐵 ∨_ℋ 𝑝)
22	2, 1, 21	mdsymlem7 31926	. . 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 458	. 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 310	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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ⊆ wss 3949 class class class wbr 5149 (class class class)co 7412 C_ℋ cch 30446 ∨_ℋ chj 30450 0_ℋc0h 30452 HAtomscat 30482 𝑀_ℋ^* cdmd 30484

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cc 10433 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193 ax-hilex 30516 ax-hfvadd 30517 ax-hvcom 30518 ax-hvass 30519 ax-hv0cl 30520 ax-hvaddid 30521 ax-hfvmul 30522 ax-hvmulid 30523 ax-hvmulass 30524 ax-hvdistr1 30525 ax-hvdistr2 30526 ax-hvmul0 30527 ax-hfi 30596 ax-his1 30599 ax-his2 30600 ax-his3 30601 ax-his4 30602 ax-hcompl 30719

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-oadd 8473 df-omul 8474 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-acn 9940 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-cn 22952 df-cnp 22953 df-lm 22954 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cfil 25004 df-cau 25005 df-cmet 25006 df-grpo 30010 df-gid 30011 df-ginv 30012 df-gdiv 30013 df-ablo 30062 df-vc 30076 df-nv 30109 df-va 30112 df-ba 30113 df-sm 30114 df-0v 30115 df-vs 30116 df-nmcv 30117 df-ims 30118 df-dip 30218 df-ssp 30239 df-ph 30330 df-cbn 30380 df-hnorm 30485 df-hba 30486 df-hvsub 30488 df-hlim 30489 df-hcau 30490 df-sh 30724 df-ch 30738 df-oc 30769 df-ch0 30770 df-shs 30825 df-span 30826 df-chj 30827 df-chsup 30828 df-pjh 30912 df-cv 31796 df-dmd 31798 df-at 31855

This theorem is referenced by: mdsymi 31928

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