dihopellsm - Mathbox for Norm Megill

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Theorem dihopellsm 41238

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 2735	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8	3, 4, 5, 6, 7	dihlss 41233	. . . 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 41233	. . . 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 2735	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
14		dihopellsm.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
15	4, 6, 13, 7, 14	dvhopellsm 41100	. . 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 1370	. 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 41236	. . . . . 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 41236	. . . . . 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 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		dihopellsm.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑣 ∈ 𝐸, 𝑤 ∈ 𝐸 ↦ (𝑖 ∈ 𝑇 ↦ ((𝑣‘𝑖) ∘ (𝑤‘𝑖))))
32	4, 17, 18, 31, 6, 13	dvhopvadd2 41077	. . . . . . . 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 1370	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩) = ⟨(𝑔 ∘ ℎ), (𝑡𝐴𝑢)⟩)
34	33	eqeq2d 2746	. . . . . 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 3482	. . . . . . . 8 ⊢ 𝑔 ∈ V
36		vex 3482	. . . . . . . 8 ⊢ ℎ ∈ V
37	35, 36	coex 7953	. . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V
38		ovex 7464	. . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V
39	37, 38	opth2 5491	. . . . . 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 1924	. 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 1537 ∃wex 1776 ∈ wcel 2106 ⟨cop 4637 ↦ cmpt 5231 ∘ ccom 5693 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17245 +_gcplusg 17298 LSSumclsm 19667 LSubSpclss 20947 HLchlt 39332 LHypclh 39967 LTrncltrn 40084 TEndoctendo 40735 DVecHcdvh 41061 DIsoHcdih 41211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tendo 40738 df-edring 40740 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212

This theorem is referenced by: dihjatcclem4 41404

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