xihopellsmN - Mathbox for Norm Megill

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Theorem xihopellsmN 39005

Description: Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
xihopellsm.b	⊢ 𝐵 = (Base‘𝐾)
xihopellsm.h	⊢ 𝐻 = (LHyp‘𝐾)
xihopellsm.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
xihopellsm.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
xihopellsm.a	⊢ 𝐴 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
xihopellsm.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
xihopellsm.l	⊢ 𝐿 = (LSubSp‘𝑈)
xihopellsm.p	⊢ ⊕ = (LSSum‘𝑈)
xihopellsm.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
xihopellsm.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
xihopellsm.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
xihopellsm.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
xihopellsmN	⊢ (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))))

Distinct variable groups: 𝑡,𝑠,𝐸 𝑔,ℎ,𝑡,𝑢,𝐹 𝑓,𝑔,𝑡,𝐻 𝑔,𝐼,ℎ,𝑡,𝑢 𝑓,𝑠,𝐾,𝑔,𝑡 𝑆,𝑔,ℎ,𝑡,𝑢 𝑈,𝑔,ℎ,𝑡,𝑢 𝑓,𝑊,𝑔,𝑠,𝑡 𝑔,𝑋,ℎ,𝑡,𝑢 𝑔,𝑌,ℎ,𝑡,𝑢 𝜑,𝑔,ℎ,𝑡,𝑢 Allowed substitution hints: 𝜑(𝑓,𝑠) 𝐴(𝑢,𝑡,𝑓,𝑔,ℎ,𝑠) 𝐵(𝑢,𝑡,𝑓,𝑔,ℎ,𝑠) ⊕ (𝑢,𝑡,𝑓,𝑔,ℎ,𝑠) 𝑆(𝑓,𝑠) 𝑇(𝑢,𝑡,𝑓,𝑔,ℎ,𝑠) 𝑈(𝑓,𝑠) 𝐸(𝑢,𝑓,𝑔,ℎ) 𝐹(𝑓,𝑠) 𝐻(𝑢,ℎ,𝑠) 𝐼(𝑓,𝑠) 𝐾(𝑢,ℎ) 𝐿(𝑢,𝑡,𝑓,𝑔,ℎ,𝑠) 𝑊(𝑢,ℎ) 𝑋(𝑓,𝑠) 𝑌(𝑓,𝑠)

**Proof of Theorem xihopellsmN**
Step	Hyp	Ref	Expression
1		xihopellsm.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		xihopellsm.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		xihopellsm.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
4		xihopellsm.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		xihopellsm.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
6		xihopellsm.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2737	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8	3, 4, 5, 6, 7	dihlss 39001	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
9	1, 2, 8	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
10		xihopellsm.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11	3, 4, 5, 6, 7	dihlss 39001	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
12	1, 10, 11	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
13		eqid 2737	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
14		xihopellsm.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
15	4, 6, 13, 7, 14	dvhopellsm 38868	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑋) ∈ (LSubSp‘𝑈) ∧ (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩))))
16	1, 9, 12, 15	syl3anc 1373	. 2 ⊢ (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩))))
17		xihopellsm.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
18		xihopellsm.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
19	1	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20	2	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → 𝑋 ∈ 𝐵)
21		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋))
22	3, 4, 17, 18, 5, 19, 20, 21	dihopcl 39004	. . . . . 6 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸))
23	1	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
24	10	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → 𝑌 ∈ 𝐵)
25		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌))
26	3, 4, 17, 18, 5, 23, 24, 25	dihopcl 39004	. . . . . 6 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))
27	22, 26	anim12dan 622	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌))) → ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)))
28	1	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸))
30		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))
31		xihopellsm.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
32	4, 17, 18, 31, 6, 13	dvhopvadd2 38845	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)) → (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩) = ⟨(𝑔 ∘ ℎ), (𝑡𝐴𝑢)⟩)
33	28, 29, 30, 32	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩) = ⟨(𝑔 ∘ ℎ), (𝑡𝐴𝑢)⟩)
34	33	eqeq2d 2748	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (⟨𝐹, 𝑆⟩ = (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩) ↔ ⟨𝐹, 𝑆⟩ = ⟨(𝑔 ∘ ℎ), (𝑡𝐴𝑢)⟩))
35		vex 3412	. . . . . . . 8 ⊢ 𝑔 ∈ V
36		vex 3412	. . . . . . . 8 ⊢ ℎ ∈ V
37	35, 36	coex 7708	. . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V
38		ovex 7246	. . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V
39	37, 38	opth2 5364	. . . . . 6 ⊢ (⟨𝐹, 𝑆⟩ = ⟨(𝑔 ∘ ℎ), (𝑡𝐴𝑢)⟩ ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))
40	34, 39	bitrdi 290	. . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (⟨𝐹, 𝑆⟩ = (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩) ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))))
41	27, 40	syldan 594	. . . 4 ⊢ ((𝜑 ∧ (⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩) ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢))))
42	41	pm5.32da 582	. . 3 ⊢ (𝜑 → (((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩)) ↔ ((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))))
43	42	4exbidv 1934	. 2 ⊢ (𝜑 → (∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))))
44	16, 43	bitrd 282	1 ⊢ (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ↔ ∃𝑔∃𝑡∃ℎ∃𝑢((⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) ∧ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ⟨cop 4547 ↦ cmpt 5135 ∘ ccom 5555 ‘cfv 6380 (class class class)co 7213 ∈ cmpo 7215 Basecbs 16760 +_gcplusg 16802 LSSumclsm 19023 LSubSpclss 19968 HLchlt 37101 LHypclh 37735 LTrncltrn 37852 TEndoctendo 38503 DVecHcdvh 38829 DIsoHcdih 38979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-riotaBAD 36704

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-undef 8015 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-0g 16946 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cntz 18711 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lvec 20140 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-llines 37249 df-lplanes 37250 df-lvols 37251 df-lines 37252 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-lhyp 37739 df-laut 37740 df-ldil 37855 df-ltrn 37856 df-trl 37910 df-tendo 38506 df-edring 38508 df-disoa 38780 df-dvech 38830 df-dib 38890 df-dic 38924 df-dih 38980

This theorem is referenced by: (None)

W3C validator