erngdvlem3-rN - Mathbox for Norm Megill

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Theorem erngdvlem3-rN 40526

Description: Lemma for eringring 40520. (Contributed by NM, 6-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)))
erngrnglem.m-r	⊢ 𝑀 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ 𝑎))

**Assertion**
Ref	Expression
erngdvlem3-rN	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Ring)

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) 𝑀(𝑓,𝑎,𝑏) 𝑂(𝑓,𝑎,𝑏)

Proof of Theorem erngdvlem3-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 2725	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
6	1, 2, 3, 4, 5	erngbase-rN 40337	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
7	6	eqcomd 2731	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐸 = (Base‘𝐷))
8		ernggrplem.p-r	. . 3 ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))
9		eqid 2725	. . . 4 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
10	1, 2, 3, 4, 9	erngfplus-rN 40338	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))))
11	8, 10	eqtr4id 2784	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+_g‘𝐷))
12		erngrnglem.m-r	. . 3 ⊢ 𝑀 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ 𝑎))
13		eqid 2725	. . . 4 ⊢ (._r‘𝐷) = (._r‘𝐷)
14	1, 2, 3, 4, 13	erngfmul-rN 40341	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (._r‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ 𝑎)))
15	12, 14	eqtr4id 2784	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑀 = (._r‘𝐷))
16		ernggrplem.b-r	. . 3 ⊢ 𝐵 = (Base‘𝐾)
17		ernggrplem.o-r	. . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
18		ernggrplem.i-r	. . 3 ⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ^◡(𝑎‘𝑓)))
19	1, 4, 16, 2, 3, 8, 17, 18	erngdvlem1-rN 40524	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
20	15	oveqd 7432	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀𝑡) = (𝑠(._r‘𝐷)𝑡))
21	20	3ad2ant1 1130	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑀𝑡) = (𝑠(._r‘𝐷)𝑡))
22	1, 2, 3, 4, 13	erngmul-rN 40342	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸)) → (𝑠(._r‘𝐷)𝑡) = (𝑡 ∘ 𝑠))
23	22	3impb 1112	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠(._r‘𝐷)𝑡) = (𝑡 ∘ 𝑠))
24	21, 23	eqtrd 2765	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑀𝑡) = (𝑡 ∘ 𝑠))
25	1, 3	tendococl 40300	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸) → (𝑡 ∘ 𝑠) ∈ 𝐸)
26	25	3com23 1123	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑡 ∘ 𝑠) ∈ 𝐸)
27	24, 26	eqeltrd 2825	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑀𝑡) ∈ 𝐸)
28	15	oveqdr 7443	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑀𝑢) = (𝑡(._r‘𝐷)𝑢))
29	1, 2, 3, 4, 13	erngmul-rN 40342	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡(._r‘𝐷)𝑢) = (𝑢 ∘ 𝑡))
30	29	3adantr1 1166	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡(._r‘𝐷)𝑢) = (𝑢 ∘ 𝑡))
31	28, 30	eqtrd 2765	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑀𝑢) = (𝑢 ∘ 𝑡))
32	31	coeq1d 5858	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑡𝑀𝑢) ∘ 𝑠) = ((𝑢 ∘ 𝑡) ∘ 𝑠))
33	15	oveqd 7432	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀(𝑡𝑀𝑢)) = (𝑠(._r‘𝐷)(𝑡𝑀𝑢)))
34	33	adantr 479	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑀𝑢)) = (𝑠(._r‘𝐷)(𝑡𝑀𝑢)))
35		simpl 481	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
36		simpr1 1191	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑠 ∈ 𝐸)
37		simpr3 1193	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑢 ∈ 𝐸)
38		simpr2 1192	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑡 ∈ 𝐸)
39	1, 3	tendococl 40300	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑢 ∘ 𝑡) ∈ 𝐸)
40	35, 37, 38, 39	syl3anc 1368	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑢 ∘ 𝑡) ∈ 𝐸)
41	31, 40	eqeltrd 2825	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑀𝑢) ∈ 𝐸)
42	1, 2, 3, 4, 13	erngmul-rN 40342	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (𝑡𝑀𝑢) ∈ 𝐸)) → (𝑠(._r‘𝐷)(𝑡𝑀𝑢)) = ((𝑡𝑀𝑢) ∘ 𝑠))
43	35, 36, 41, 42	syl12anc 835	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(._r‘𝐷)(𝑡𝑀𝑢)) = ((𝑡𝑀𝑢) ∘ 𝑠))
44	34, 43	eqtrd 2765	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑀𝑢)) = ((𝑡𝑀𝑢) ∘ 𝑠))
45	15	oveqd 7432	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑠𝑀𝑡)𝑀𝑢) = ((𝑠𝑀𝑡)(._r‘𝐷)𝑢))
46	45	adantr 479	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = ((𝑠𝑀𝑡)(._r‘𝐷)𝑢))
47	27	3adant3r3 1181	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑡) ∈ 𝐸)
48	1, 2, 3, 4, 13	erngmul-rN 40342	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠𝑀𝑡) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)(._r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑀𝑡)))
49	35, 47, 37, 48	syl12anc 835	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)(._r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑀𝑡)))
50	15	oveqdr 7443	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑡) = (𝑠(._r‘𝐷)𝑡))
51	22	3adantr3 1168	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(._r‘𝐷)𝑡) = (𝑡 ∘ 𝑠))
52	50, 51	eqtrd 2765	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑡) = (𝑡 ∘ 𝑠))
53	52	coeq2d 5859	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑢 ∘ (𝑠𝑀𝑡)) = (𝑢 ∘ (𝑡 ∘ 𝑠)))
54	46, 49, 53	3eqtrd 2769	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = (𝑢 ∘ (𝑡 ∘ 𝑠)))
55		coass 6264	. . . 4 ⊢ ((𝑢 ∘ 𝑡) ∘ 𝑠) = (𝑢 ∘ (𝑡 ∘ 𝑠))
56	54, 55	eqtr4di 2783	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = ((𝑢 ∘ 𝑡) ∘ 𝑠))
57	32, 44, 56	3eqtr4rd 2776	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑀𝑢) = (𝑠𝑀(𝑡𝑀𝑢)))
58	1, 2, 3, 8	tendodi2 40313	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸)) → ((𝑡𝑃𝑢) ∘ 𝑠) = ((𝑡 ∘ 𝑠)𝑃(𝑢 ∘ 𝑠)))
59	35, 38, 37, 36, 58	syl13anc 1369	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑡𝑃𝑢) ∘ 𝑠) = ((𝑡 ∘ 𝑠)𝑃(𝑢 ∘ 𝑠)))
60	15	oveqd 7432	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀(𝑡𝑃𝑢)) = (𝑠(._r‘𝐷)(𝑡𝑃𝑢)))
61	60	adantr 479	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑃𝑢)) = (𝑠(._r‘𝐷)(𝑡𝑃𝑢)))
62	1, 2, 3, 8	tendoplcl 40309	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸) → (𝑡𝑃𝑢) ∈ 𝐸)
63	35, 38, 37, 62	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑡𝑃𝑢) ∈ 𝐸)
64	1, 2, 3, 4, 13	erngmul-rN 40342	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ (𝑡𝑃𝑢) ∈ 𝐸)) → (𝑠(._r‘𝐷)(𝑡𝑃𝑢)) = ((𝑡𝑃𝑢) ∘ 𝑠))
65	35, 36, 63, 64	syl12anc 835	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(._r‘𝐷)(𝑡𝑃𝑢)) = ((𝑡𝑃𝑢) ∘ 𝑠))
66	61, 65	eqtrd 2765	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑃𝑢)) = ((𝑡𝑃𝑢) ∘ 𝑠))
67	15	oveqdr 7443	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑢) = (𝑠(._r‘𝐷)𝑢))
68	1, 2, 3, 4, 13	erngmul-rN 40342	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(._r‘𝐷)𝑢) = (𝑢 ∘ 𝑠))
69	68	3adantr2 1167	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠(._r‘𝐷)𝑢) = (𝑢 ∘ 𝑠))
70	67, 69	eqtrd 2765	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀𝑢) = (𝑢 ∘ 𝑠))
71	52, 70	oveq12d 7433	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑡)𝑃(𝑠𝑀𝑢)) = ((𝑡 ∘ 𝑠)𝑃(𝑢 ∘ 𝑠)))
72	59, 66, 71	3eqtr4d 2775	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑀(𝑡𝑃𝑢)) = ((𝑠𝑀𝑡)𝑃(𝑠𝑀𝑢)))
73	1, 2, 3, 8	tendodi1 40312	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑢 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸)) → (𝑢 ∘ (𝑠𝑃𝑡)) = ((𝑢 ∘ 𝑠)𝑃(𝑢 ∘ 𝑡)))
74	35, 37, 36, 38, 73	syl13anc 1369	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑢 ∘ (𝑠𝑃𝑡)) = ((𝑢 ∘ 𝑠)𝑃(𝑢 ∘ 𝑡)))
75	15	adantr 479	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → 𝑀 = (._r‘𝐷))
76	75	oveqd 7432	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑀𝑢) = ((𝑠𝑃𝑡)(._r‘𝐷)𝑢))
77	1, 2, 3, 8	tendoplcl 40309	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸) → (𝑠𝑃𝑡) ∈ 𝐸)
78	77	3adant3r3 1181	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (𝑠𝑃𝑡) ∈ 𝐸)
79	1, 2, 3, 4, 13	erngmul-rN 40342	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠𝑃𝑡) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)(._r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑃𝑡)))
80	35, 78, 37, 79	syl12anc 835	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)(._r‘𝐷)𝑢) = (𝑢 ∘ (𝑠𝑃𝑡)))
81	76, 80	eqtrd 2765	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑀𝑢) = (𝑢 ∘ (𝑠𝑃𝑡)))
82	70, 31	oveq12d 7433	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑀𝑢)𝑃(𝑡𝑀𝑢)) = ((𝑢 ∘ 𝑠)𝑃(𝑢 ∘ 𝑡)))
83	74, 81, 82	3eqtr4d 2775	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → ((𝑠𝑃𝑡)𝑀𝑢) = ((𝑠𝑀𝑢)𝑃(𝑡𝑀𝑢)))
84	1, 2, 3	tendoidcl 40297	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
85	15	oveqd 7432	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝑇)𝑀𝑠) = (( I ↾ 𝑇)(._r‘𝐷)𝑠))
86	85	adantr 479	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇)𝑀𝑠) = (( I ↾ 𝑇)(._r‘𝐷)𝑠))
87		simpl 481	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
88	84	adantr 479	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ( I ↾ 𝑇) ∈ 𝐸)
89		simpr 483	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → 𝑠 ∈ 𝐸)
90	1, 2, 3, 4, 13	erngmul-rN 40342	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸 ∧ 𝑠 ∈ 𝐸)) → (( I ↾ 𝑇)(._r‘𝐷)𝑠) = (𝑠 ∘ ( I ↾ 𝑇)))
91	87, 88, 89, 90	syl12anc 835	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇)(._r‘𝐷)𝑠) = (𝑠 ∘ ( I ↾ 𝑇)))
92	1, 2, 3	tendo1mulr 40299	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠 ∘ ( I ↾ 𝑇)) = 𝑠)
93	86, 91, 92	3eqtrd 2769	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇)𝑀𝑠) = 𝑠)
94	15	oveqd 7432	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀( I ↾ 𝑇)) = (𝑠(._r‘𝐷)( I ↾ 𝑇)))
95	94	adantr 479	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠𝑀( I ↾ 𝑇)) = (𝑠(._r‘𝐷)( I ↾ 𝑇)))
96	1, 2, 3, 4, 13	erngmul-rN 40342	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑠(._r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ 𝑠))
97	87, 89, 88, 96	syl12anc 835	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠(._r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ 𝑠))
98	1, 2, 3	tendo1mul 40298	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑠) = 𝑠)
99	95, 97, 98	3eqtrd 2769	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠𝑀( I ↾ 𝑇)) = 𝑠)
100	7, 11, 15, 19, 27, 57, 72, 83, 84, 93, 99	isringd 20229	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Ring)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5226 I cid 5569 ^◡ccnv 5671 ↾ cres 5674 ∘ ccom 5676 ‘cfv 6542 (class class class)co 7415 ∈ cmpo 7417 Basecbs 17177 +_gcplusg 17230 ._rcmulr 17231 Ringcrg 20175 HLchlt 38877 LHypclh 39512 LTrncltrn 39629 TEndoctendo 40280 EDRing_Rcedring-rN 40282

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-riotaBAD 38480

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-undef 8275 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-mulr 17244 df-0g 17420 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-mgp 20077 df-ring 20177 df-oposet 38703 df-ol 38705 df-oml 38706 df-covers 38793 df-ats 38794 df-atl 38825 df-cvlat 38849 df-hlat 38878 df-llines 39026 df-lplanes 39027 df-lvols 39028 df-lines 39029 df-psubsp 39031 df-pmap 39032 df-padd 39324 df-lhyp 39516 df-laut 39517 df-ldil 39632 df-ltrn 39633 df-trl 39687 df-tendo 40283 df-edring-rN 40284

This theorem is referenced by: erngdvlem4-rN 40527 erngring-rN 40528

W3C validator