erngdvlem1-rN - Mathbox for Norm Megill

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Theorem erngdvlem1-rN 35302

Description: Lemma for eringring 35298. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ernggrp.h-r	⊢ 𝐻 = (LHyp‘𝐾)
ernggrp.d-r	⊢ 𝐷 = ((EDRing_R‘𝐾)‘𝑊)
ernggrplem.b-r	⊢ 𝐵 = (Base‘𝐾)
ernggrplem.t-r	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
ernggrplem.e-r	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
ernggrplem.p-r	⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))
ernggrplem.o-r	⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
ernggrplem.i-r	⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ^◡(𝑎‘𝑓)))

**Assertion**
Ref	Expression
erngdvlem1-rN	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)

Distinct variable groups: 𝐵,𝑓 𝑎,𝑏,𝐸 𝑓,𝑎,𝐾,𝑏 𝑓,𝐻 𝑇,𝑎,𝑏,𝑓 𝑊,𝑎,𝑏,𝑓 Allowed substitution hints: 𝐵(𝑎,𝑏) 𝐷(𝑓,𝑎,𝑏) 𝑃(𝑓,𝑎,𝑏) 𝐸(𝑓) 𝐻(𝑎,𝑏) 𝐼(𝑓,𝑎,𝑏) 𝑂(𝑓,𝑎,𝑏)

Proof of Theorem erngdvlem1-rN Dummy variables 𝑡 𝑠 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ernggrp.h-r	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		ernggrplem.t-r	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		ernggrplem.e-r	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4		ernggrp.d-r	. . . 4 ⊢ 𝐷 = ((EDRing_R‘𝐾)‘𝑊)
5		eqid 2610	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
6	1, 2, 3, 4, 5	erngbase-rN 35115	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
7	6	eqcomd 2616	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐸 = (Base‘𝐷))
8		eqid 2610	. . . 4 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
9	1, 2, 3, 4, 8	erngfplus-rN 35116	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))))
10		ernggrplem.p-r	. . 3 ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))
11	9, 10	syl6reqr 2663	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+_g‘𝐷))
12	1, 2, 3, 10	tendoplcl 35087	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) ∈ 𝐸)
13	1, 2, 3, 10	tendoplass 35089	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑃𝑢) = (𝑠𝑃(𝑡𝑃𝑢)))
14		ernggrplem.b-r	. . 3 ⊢ 𝐵 = (Base‘𝐾)
15		ernggrplem.o-r	. . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
16	14, 1, 2, 3, 15	tendo0cl 35096	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸)
17	14, 1, 2, 3, 15, 10	tendo0pl 35097	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑂𝑃𝑠) = 𝑠)
18		ernggrplem.i-r	. . 3 ⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ^◡(𝑎‘𝑓)))
19	1, 2, 3, 18	tendoicl 35102	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝐼‘𝑠) ∈ 𝐸)
20	1, 2, 3, 18, 14, 10, 15	tendoipl 35103	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ((𝐼‘𝑠)𝑃𝑠) = 𝑂)
21	7, 11, 12, 13, 16, 17, 19, 20	isgrpd 17267	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 I cid 4948 ^◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 ‘cfv 5804 ↦ cmpt2 6551 Basecbs 15695 +_gcplusg 15768 Grpcgrp 17245 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 TEndoctendo 35058 EDRing_Rcedring-rN 35060

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-0g 15925 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tendo 35061 df-edring-rN 35062

This theorem is referenced by: erngdvlem2-rN 35303 erngdvlem3-rN 35304 erngdvlem4-rN 35305

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