xihopellsmN - Mathbox for Norm Megill

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Theorem xihopellsmN 41242

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 2729	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8	3, 4, 5, 6, 7	dihlss 41238	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
9	1, 2, 8	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
10		xihopellsm.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11	3, 4, 5, 6, 7	dihlss 41238	. . . 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 2729	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
14		xihopellsm.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
15	4, 6, 13, 7, 14	dvhopellsm 41105	. . 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 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 41241	. . . . . 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 41241	. . . . . 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		xihopellsm.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
32	4, 17, 18, 31, 6, 13	dvhopvadd2 41082	. . . . . . . 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 2740	. . . . . 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 3448	. . . . . . . 8 ⊢ 𝑔 ∈ V
36		vex 3448	. . . . . . . 8 ⊢ ℎ ∈ V
37	35, 36	coex 7886	. . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V
38		ovex 7402	. . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V
39	37, 38	opth2 5435	. . . . . 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 1926	. 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 1540 ∃wex 1779 ∈ wcel 2109 ⟨cop 4591 ↦ cmpt 5183 ∘ ccom 5635 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 Basecbs 17156 +_gcplusg 17197 LSSumclsm 19549 LSubSpclss 20870 HLchlt 39337 LHypclh 39972 LTrncltrn 40089 TEndoctendo 40740 DVecHcdvh 41066 DIsoHcdih 41216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11102 ax-resscn 11103 ax-1cn 11104 ax-icn 11105 ax-addcl 11106 ax-addrcl 11107 ax-mulcl 11108 ax-mulrcl 11109 ax-mulcom 11110 ax-addass 11111 ax-mulass 11112 ax-distr 11113 ax-i2m1 11114 ax-1ne0 11115 ax-1rid 11116 ax-rnegex 11117 ax-rrecex 11118 ax-cnre 11119 ax-pre-lttri 11120 ax-pre-lttrn 11121 ax-pre-ltadd 11122 ax-pre-mulgt0 11123 ax-riotaBAD 38940

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-undef 8229 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11385 df-neg 11386 df-nn 12165 df-2 12227 df-3 12228 df-4 12229 df-5 12230 df-6 12231 df-n0 12421 df-z 12508 df-uz 12772 df-fz 13447 df-struct 17094 df-sets 17111 df-slot 17129 df-ndx 17141 df-base 17157 df-ress 17178 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-0g 17381 df-proset 18236 df-poset 18255 df-plt 18270 df-lub 18286 df-glb 18287 df-join 18288 df-meet 18289 df-p0 18365 df-p1 18366 df-lat 18374 df-clat 18441 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-grp 18851 df-minusg 18852 df-sbg 18853 df-subg 19038 df-cntz 19232 df-lsm 19551 df-cmn 19697 df-abl 19698 df-mgp 20062 df-rng 20074 df-ur 20103 df-ring 20156 df-oppr 20258 df-dvdsr 20278 df-unit 20279 df-invr 20309 df-dvr 20322 df-drng 20652 df-lmod 20801 df-lss 20871 df-lsp 20911 df-lvec 21043 df-oposet 39163 df-ol 39165 df-oml 39166 df-covers 39253 df-ats 39254 df-atl 39285 df-cvlat 39309 df-hlat 39338 df-llines 39486 df-lplanes 39487 df-lvols 39488 df-lines 39489 df-psubsp 39491 df-pmap 39492 df-padd 39784 df-lhyp 39976 df-laut 39977 df-ldil 40092 df-ltrn 40093 df-trl 40147 df-tendo 40743 df-edring 40745 df-disoa 41017 df-dvech 41067 df-dib 41127 df-dic 41161 df-dih 41217

This theorem is referenced by: (None)

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