xihopellsmN - Mathbox for Norm Megill

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Theorem xihopellsmN 41761

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 2741	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
8	3, 4, 5, 6, 7	dihlss 41757	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
9	1, 2, 8	syl2anc 591	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
10		xihopellsm.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
11	3, 4, 5, 6, 7	dihlss 41757	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
12	1, 10, 11	syl2anc 591	. . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
13		eqid 2741	. . . 4 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
14		xihopellsm.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
15	4, 6, 13, 7, 14	dvhopellsm 41624	. . 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 1380	. 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20	2	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → 𝑋 ∈ 𝐵)
21		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋))
22	3, 4, 17, 18, 5, 19, 20, 21	dihopcl 41760	. . . . . 6 ⊢ ((𝜑 ∧ ⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋)) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸))
23	1	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
24	10	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → 𝑌 ∈ 𝐵)
25		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌))
26	3, 4, 17, 18, 5, 23, 24, 25	dihopcl 41760	. . . . . 6 ⊢ ((𝜑 ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌)) → (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))
27	22, 26	anim12dan 626	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑔, 𝑡⟩ ∈ (𝐼‘𝑋) ∧ ⟨ℎ, 𝑢⟩ ∈ (𝐼‘𝑌))) → ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸)))
28	1	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simprl 777	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸))
30		simprr 779	. . . . . . . 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 41601	. . . . . . . 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 1380	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ (ℎ ∈ 𝑇 ∧ 𝑢 ∈ 𝐸))) → (⟨𝑔, 𝑡⟩(+_g‘𝑈)⟨ℎ, 𝑢⟩) = ⟨(𝑔 ∘ ℎ), (𝑡𝐴𝑢)⟩)
34	33	eqeq2d 2752	. . . . . 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 3437	. . . . . . . 8 ⊢ 𝑔 ∈ V
36		vex 3437	. . . . . . . 8 ⊢ ℎ ∈ V
37	35, 36	coex 7874	. . . . . . 7 ⊢ (𝑔 ∘ ℎ) ∈ V
38		ovex 7393	. . . . . . 7 ⊢ (𝑡𝐴𝑢) ∈ V
39	37, 38	opth2 5423	. . . . . 6 ⊢ (⟨𝐹, 𝑆⟩ = ⟨(𝑔 ∘ ℎ), (𝑡𝐴𝑢)⟩ ↔ (𝐹 = (𝑔 ∘ ℎ) ∧ 𝑆 = (𝑡𝐴𝑢)))
40	34, 39	bitrdi 289	. . . . 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 598	. . . 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 585	. . 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 281	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 208 ∧ wa 397 = wceq 1548 ∃wex 1787 ∈ wcel 2121 ⟨cop 4564 ↦ cmpt 5156 ∘ ccom 5625 ‘cfv 6489 (class class class)co 7360 ∈ cmpo 7362 Basecbs 17174 +_gcplusg 17215 LSSumclsm 19604 LSubSpclss 20925 HLchlt 39857 LHypclh 40491 LTrncltrn 40608 TEndoctendo 41259 DVecHcdvh 41585 DIsoHcdih 41735

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 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-riotaBAD 39460

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-0g 17399 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18393 df-clat 18460 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19094 df-cntz 19287 df-lsm 19606 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-drng 20707 df-lmod 20856 df-lss 20926 df-lsp 20966 df-lvec 21097 df-oposet 39683 df-ol 39685 df-oml 39686 df-covers 39773 df-ats 39774 df-atl 39805 df-cvlat 39829 df-hlat 39858 df-llines 40005 df-lplanes 40006 df-lvols 40007 df-lines 40008 df-psubsp 40010 df-pmap 40011 df-padd 40303 df-lhyp 40495 df-laut 40496 df-ldil 40611 df-ltrn 40612 df-trl 40666 df-tendo 41262 df-edring 41264 df-disoa 41536 df-dvech 41586 df-dib 41646 df-dic 41680 df-dih 41736

This theorem is referenced by: (None)

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