dihopellsm - Mathbox for Norm Megill

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Theorem dihopellsm 41454

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

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

**Assertion**
Ref	Expression
dihopellsm	⊢ (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: 𝑤,𝑣,𝐸 𝑔,ℎ,𝑡,𝑢,𝐹 𝑔,𝑖,𝐻,𝑡 𝑔,𝐼,ℎ,𝑡,𝑢 𝑣,𝑔,𝑤,𝐾,𝑖,𝑡 𝑆,𝑔,ℎ,𝑡,𝑢 𝑈,𝑔,ℎ,𝑡,𝑢 𝑔,𝑊,𝑖,𝑡,𝑣,𝑤 𝑔,𝑋,ℎ,𝑡,𝑢 𝑔,𝑌,ℎ,𝑡,𝑢 𝜑,𝑔,ℎ,𝑡,𝑢 Allowed substitution hints: 𝜑(𝑤,𝑣,𝑖) 𝐴(𝑤,𝑣,𝑢,𝑡,𝑔,ℎ,𝑖) 𝐵(𝑤,𝑣,𝑢,𝑡,𝑔,ℎ,𝑖) ⊕ (𝑤,𝑣,𝑢,𝑡,𝑔,ℎ,𝑖) 𝑆(𝑤,𝑣,𝑖) 𝑇(𝑤,𝑣,𝑢,𝑡,𝑔,ℎ,𝑖) 𝑈(𝑤,𝑣,𝑖) 𝐸(𝑢,𝑡,𝑔,ℎ,𝑖) 𝐹(𝑤,𝑣,𝑖) 𝐻(𝑤,𝑣,𝑢,ℎ) 𝐼(𝑤,𝑣,𝑖) 𝐾(𝑢,ℎ) 𝐿(𝑤,𝑣,𝑢,𝑡,𝑔,ℎ,𝑖) 𝑊(𝑢,ℎ) 𝑋(𝑤,𝑣,𝑖) 𝑌(𝑤,𝑣,𝑖)

**Proof of Theorem dihopellsm**
Step	Hyp	Ref	Expression
1		dihopellsm.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dihopellsm.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		dihopellsm.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
4		dihopellsm.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		dihopellsm.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
6		dihopellsm.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		eqid 2734	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8	3, 4, 5, 6, 7	dihlss 41449	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
9	1, 2, 8	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
10		dihopellsm.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11	3, 4, 5, 6, 7	dihlss 41449	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
12	1, 10, 11	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
13		eqid 2734	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
14		dihopellsm.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
15	4, 6, 13, 7, 14	dvhopellsm 41316	. . 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		dihopellsm.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
18		dihopellsm.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
19	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → 𝑋 ∈ 𝐵)
21		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋))
22	3, 4, 17, 18, 5, 19, 20, 21	dihopcl 41452	. . . . . 6 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸))
23	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
24	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → 𝑌 ∈ 𝐵)
25		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌))
26	3, 4, 17, 18, 5, 23, 24, 25	dihopcl 41452	. . . . . 6 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))
27	22, 26	anim12dan 619	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌))) → ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)))
28	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸))
30		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))
31		dihopellsm.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑣 ∈ 𝐸, 𝑤 ∈ 𝐸 ↦ (𝑖 ∈ 𝑇 ↦ ((𝑣‘𝑖) ∘ (𝑤‘𝑖))))
32	4, 17, 18, 31, 6, 13	dvhopvadd2 41293	. . . . . . . 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 2745	. . . . . 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 3442	. . . . . . . 8 ⊢ 𝑔 ∈ V
36		vex 3442	. . . . . . . 8 ⊢ ℎ ∈ V
37	35, 36	coex 7870	. . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V
38		ovex 7389	. . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V
39	37, 38	opth2 5426	. . . . . 6 ⊢ (⟨𝐹, 𝑆⟩ = ⟨(𝑔 ∘ ℎ), (𝑡𝐴𝑢)⟩ ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))
40	34, 39	bitrdi 287	. . . . 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 591	. . . 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 579	. . 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 1927	. 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 279	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 206 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ⟨cop 4584 ↦ cmpt 5177 ∘ ccom 5626 ‘cfv 6490 (class class class)co 7356 ∈ cmpo 7358 Basecbs 17134 +_gcplusg 17175 LSSumclsm 19561 LSubSpclss 20880 HLchlt 39549 LHypclh 40183 LTrncltrn 40300 TEndoctendo 40951 DVecHcdvh 41277 DIsoHcdih 41427

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-riotaBAD 39152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-0g 17359 df-proset 18215 df-poset 18234 df-plt 18249 df-lub 18265 df-glb 18266 df-join 18267 df-meet 18268 df-p0 18344 df-p1 18345 df-lat 18353 df-clat 18420 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-cntz 19244 df-lsm 19563 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20271 df-dvdsr 20291 df-unit 20292 df-invr 20322 df-dvr 20335 df-drng 20662 df-lmod 20811 df-lss 20881 df-lsp 20921 df-lvec 21053 df-oposet 39375 df-ol 39377 df-oml 39378 df-covers 39465 df-ats 39466 df-atl 39497 df-cvlat 39521 df-hlat 39550 df-llines 39697 df-lplanes 39698 df-lvols 39699 df-lines 39700 df-psubsp 39702 df-pmap 39703 df-padd 39995 df-lhyp 40187 df-laut 40188 df-ldil 40303 df-ltrn 40304 df-trl 40358 df-tendo 40954 df-edring 40956 df-disoa 41228 df-dvech 41278 df-dib 41338 df-dic 41372 df-dih 41428

This theorem is referenced by: dihjatcclem4 41620

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