erngdvlem4-rN - Mathbox for Norm Megill

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Theorem erngdvlem4-rN 40524

Description: Lemma for erngdv 40518. (Contributed by NM, 11-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))
edlemk6.j-r	⊢ ∨ = (join‘𝐾)
edlemk6.m-r	⊢ ∧ = (meet‘𝐾)
edlemk6.r-r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
edlemk6.p-r	⊢ 𝑄 = ((oc‘𝐾)‘𝑊)
edlemk6.z-r	⊢ 𝑍 = ((𝑄 ∨ (𝑅‘𝑏)) ∧ ((ℎ‘𝑄) ∨ (𝑅‘(𝑏 ∘ ^◡(𝑠‘ℎ)))))
edlemk6.y-r	⊢ 𝑌 = ((𝑄 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
edlemk6.x-r	⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘(𝑠‘ℎ)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑄) = 𝑌))
edlemk6.u-r	⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if((𝑠‘ℎ) = ℎ, 𝑔, 𝑋))

**Assertion**
Ref	Expression
erngdvlem4-rN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) → 𝐷 ∈ DivRing)

Distinct variable groups: 𝐵,𝑓 𝐷,𝑠 𝑎,𝑏,𝑠,𝐸 𝑓,𝑎,𝐾,𝑏,𝑠 𝑓,𝐻,𝑠 𝑂,𝑠 𝑇,𝑎,𝑏,𝑓,𝑠 𝑊,𝑎,𝑏,𝑓,𝑠 𝑃,𝑠 𝑔,𝑏,𝑧, ∧ ∨ ,𝑏,𝑔,𝑧 𝐵,𝑏 𝑔,𝑠,𝐵,𝑧 𝐻,𝑏,𝑔,𝑧 𝑔,𝐾,𝑧 𝑀,𝑠 𝑃,𝑔,𝑧 𝑄,𝑏,𝑔,𝑧 𝑅,𝑏,𝑔,𝑧 𝑇,𝑔,𝑧 𝑔,𝑊,𝑧 𝑧,𝑌 𝑔,𝑍 𝑓,𝑔,𝑧 ℎ,𝑏,𝑔,𝑠,𝑧 Allowed substitution hints: 𝐵(ℎ,𝑎) 𝐷(𝑧,𝑓,𝑔,ℎ,𝑎,𝑏) 𝑃(𝑓,ℎ,𝑎,𝑏) 𝑄(𝑓,ℎ,𝑠,𝑎) 𝑅(𝑓,ℎ,𝑠,𝑎) 𝑇(ℎ) 𝑈(𝑧,𝑓,𝑔,ℎ,𝑠,𝑎,𝑏) 𝐸(𝑧,𝑓,𝑔,ℎ) 𝐻(ℎ,𝑎) 𝐼(𝑧,𝑓,𝑔,ℎ,𝑠,𝑎,𝑏) ∨ (𝑓,ℎ,𝑠,𝑎) 𝐾(ℎ) 𝑀(𝑧,𝑓,𝑔,ℎ,𝑎,𝑏) ∧ (𝑓,ℎ,𝑠,𝑎) 𝑂(𝑧,𝑓,𝑔,ℎ,𝑎,𝑏) 𝑊(ℎ) 𝑋(𝑧,𝑓,𝑔,ℎ,𝑠,𝑎,𝑏) 𝑌(𝑓,𝑔,ℎ,𝑠,𝑎,𝑏) 𝑍(𝑧,𝑓,ℎ,𝑠,𝑎,𝑏)

Proof of Theorem erngdvlem4-rN Dummy variables 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ernggrp.h-r	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		ernggrplem.t-r	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		ernggrplem.e-r	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4		ernggrp.d-r	. . . . 5 ⊢ 𝐷 = ((EDRing_R‘𝐾)‘𝑊)
5		eqid 2725	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
6	1, 2, 3, 4, 5	erngbase-rN 40334	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = 𝐸)
7	6	eqcomd 2731	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐸 = (Base‘𝐷))
8	7	adantr 479	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) → 𝐸 = (Base‘𝐷))
9		erngrnglem.m-r	. . . 4 ⊢ 𝑀 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ 𝑎))
10		eqid 2725	. . . . 5 ⊢ (._r‘𝐷) = (._r‘𝐷)
11	1, 2, 3, 4, 10	erngfmul-rN 40338	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (._r‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑏 ∘ 𝑎)))
12	9, 11	eqtr4id 2784	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑀 = (._r‘𝐷))
13	12	adantr 479	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) → 𝑀 = (._r‘𝐷))
14		ernggrplem.b-r	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
15		ernggrplem.o-r	. . . . . . 7 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
16	14, 1, 2, 3, 15	tendo0cl 40315	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸)
17	16, 6	eleqtrrd 2828	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ (Base‘𝐷))
18		ernggrplem.p-r	. . . . . . . 8 ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓))))
19		eqid 2725	. . . . . . . . 9 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
20	1, 2, 3, 4, 19	erngfplus-rN 40335	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘𝐷) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑎‘𝑓) ∘ (𝑏‘𝑓)))))
21	18, 20	eqtr4id 2784	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑃 = (+_g‘𝐷))
22	21	oveqd 7430	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑂𝑃𝑂) = (𝑂(+_g‘𝐷)𝑂))
23	14, 1, 2, 3, 15, 18	tendo0pl 40316	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑂 ∈ 𝐸) → (𝑂𝑃𝑂) = 𝑂)
24	16, 23	mpdan 685	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑂𝑃𝑂) = 𝑂)
25	22, 24	eqtr3d 2767	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑂(+_g‘𝐷)𝑂) = 𝑂)
26		ernggrplem.i-r	. . . . . . 7 ⊢ 𝐼 = (𝑎 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ^◡(𝑎‘𝑓)))
27	1, 4, 14, 2, 3, 18, 15, 26	erngdvlem1-rN 40521	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Grp)
28		eqid 2725	. . . . . . 7 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
29	5, 19, 28	isgrpid2 18932	. . . . . 6 ⊢ (𝐷 ∈ Grp → ((𝑂 ∈ (Base‘𝐷) ∧ (𝑂(+_g‘𝐷)𝑂) = 𝑂) ↔ (0_g‘𝐷) = 𝑂))
30	27, 29	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑂 ∈ (Base‘𝐷) ∧ (𝑂(+_g‘𝐷)𝑂) = 𝑂) ↔ (0_g‘𝐷) = 𝑂))
31	17, 25, 30	mpbi2and 710	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘𝐷) = 𝑂)
32	31	eqcomd 2731	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 = (0_g‘𝐷))
33	32	adantr 479	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) → 𝑂 = (0_g‘𝐷))
34	1, 2, 3	tendoidcl 40294	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
35	34, 6	eleqtrrd 2828	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ (Base‘𝐷))
36	6	eleq2d 2811	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ (Base‘𝐷) ↔ 𝑢 ∈ 𝐸))
37		simpl 481	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
38	34	adantr 479	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → ( I ↾ 𝑇) ∈ 𝐸)
39		simpr 483	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → 𝑢 ∈ 𝐸)
40	1, 2, 3, 4, 10	erngmul-rN 40339	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (( I ↾ 𝑇) ∈ 𝐸 ∧ 𝑢 ∈ 𝐸)) → (( I ↾ 𝑇)(._r‘𝐷)𝑢) = (𝑢 ∘ ( I ↾ 𝑇)))
41	37, 38, 39, 40	syl12anc 835	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇)(._r‘𝐷)𝑢) = (𝑢 ∘ ( I ↾ 𝑇)))
42	1, 2, 3	tendo1mulr 40296	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢 ∘ ( I ↾ 𝑇)) = 𝑢)
43	41, 42	eqtrd 2765	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇)(._r‘𝐷)𝑢) = 𝑢)
44	1, 2, 3, 4, 10	erngmul-rN 40339	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑢 ∈ 𝐸 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑢(._r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ 𝑢))
45	37, 39, 38, 44	syl12anc 835	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢(._r‘𝐷)( I ↾ 𝑇)) = (( I ↾ 𝑇) ∘ 𝑢))
46	1, 2, 3	tendo1mul 40295	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑢) = 𝑢)
47	45, 46	eqtrd 2765	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → (𝑢(._r‘𝐷)( I ↾ 𝑇)) = 𝑢)
48	43, 47	jca 510	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑢 ∈ 𝐸) → ((( I ↾ 𝑇)(._r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(._r‘𝐷)( I ↾ 𝑇)) = 𝑢))
49	48	ex 411	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ 𝐸 → ((( I ↾ 𝑇)(._r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(._r‘𝐷)( I ↾ 𝑇)) = 𝑢)))
50	36, 49	sylbid 239	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑢 ∈ (Base‘𝐷) → ((( I ↾ 𝑇)(._r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(._r‘𝐷)( I ↾ 𝑇)) = 𝑢)))
51	50	ralrimiv 3135	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∀𝑢 ∈ (Base‘𝐷)((( I ↾ 𝑇)(._r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(._r‘𝐷)( I ↾ 𝑇)) = 𝑢))
52	1, 4, 14, 2, 3, 18, 15, 26, 9	erngdvlem3-rN 40523	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 ∈ Ring)
53		eqid 2725	. . . . . . 7 ⊢ (1_r‘𝐷) = (1_r‘𝐷)
54	5, 10, 53	isringid 20206	. . . . . 6 ⊢ (𝐷 ∈ Ring → ((( I ↾ 𝑇) ∈ (Base‘𝐷) ∧ ∀𝑢 ∈ (Base‘𝐷)((( I ↾ 𝑇)(._r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(._r‘𝐷)( I ↾ 𝑇)) = 𝑢)) ↔ (1_r‘𝐷) = ( I ↾ 𝑇)))
55	52, 54	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((( I ↾ 𝑇) ∈ (Base‘𝐷) ∧ ∀𝑢 ∈ (Base‘𝐷)((( I ↾ 𝑇)(._r‘𝐷)𝑢) = 𝑢 ∧ (𝑢(._r‘𝐷)( I ↾ 𝑇)) = 𝑢)) ↔ (1_r‘𝐷) = ( I ↾ 𝑇)))
56	35, 51, 55	mpbi2and 710	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (1_r‘𝐷) = ( I ↾ 𝑇))
57	56	eqcomd 2731	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) = (1_r‘𝐷))
58	57	adantr 479	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) → ( I ↾ 𝑇) = (1_r‘𝐷))
59	52	adantr 479	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) → 𝐷 ∈ Ring)
60		simp1l 1194	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
61	12	oveqd 7430	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀𝑡) = (𝑠(._r‘𝐷)𝑡))
62	60, 61	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → (𝑠𝑀𝑡) = (𝑠(._r‘𝐷)𝑡))
63		simp2l 1196	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → 𝑠 ∈ 𝐸)
64		simp3l 1198	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → 𝑡 ∈ 𝐸)
65	1, 2, 3, 4, 10	erngmul-rN 40339	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸)) → (𝑠(._r‘𝐷)𝑡) = (𝑡 ∘ 𝑠))
66	60, 63, 64, 65	syl12anc 835	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → (𝑠(._r‘𝐷)𝑡) = (𝑡 ∘ 𝑠))
67	62, 66	eqtrd 2765	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → (𝑠𝑀𝑡) = (𝑡 ∘ 𝑠))
68		simp3 1135	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂))
69		simp2 1134	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂))
70	14, 1, 2, 3, 15	tendoconid 40354	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝑡 ∘ 𝑠) ≠ 𝑂)
71	60, 68, 69, 70	syl3anc 1368	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → (𝑡 ∘ 𝑠) ≠ 𝑂)
72	67, 71	eqnetrd 2998	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂) ∧ (𝑡 ∈ 𝐸 ∧ 𝑡 ≠ 𝑂)) → (𝑠𝑀𝑡) ≠ 𝑂)
73	14, 1, 2, 3, 15	tendo1ne0 40353	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ 𝑂)
74	73	adantr 479	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) → ( I ↾ 𝑇) ≠ 𝑂)
75		simpll 765	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
76		simplrl 775	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → ℎ ∈ 𝑇)
77		simpr 483	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂))
78		edlemk6.j-r	. . . . 5 ⊢ ∨ = (join‘𝐾)
79		edlemk6.m-r	. . . . 5 ⊢ ∧ = (meet‘𝐾)
80		edlemk6.r-r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
81		edlemk6.p-r	. . . . 5 ⊢ 𝑄 = ((oc‘𝐾)‘𝑊)
82		edlemk6.z-r	. . . . 5 ⊢ 𝑍 = ((𝑄 ∨ (𝑅‘𝑏)) ∧ ((ℎ‘𝑄) ∨ (𝑅‘(𝑏 ∘ ^◡(𝑠‘ℎ)))))
83		edlemk6.y-r	. . . . 5 ⊢ 𝑌 = ((𝑄 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
84		edlemk6.x-r	. . . . 5 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘(𝑠‘ℎ)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑄) = 𝑌))
85		edlemk6.u-r	. . . . 5 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if((𝑠‘ℎ) = ℎ, 𝑔, 𝑋))
86	14, 78, 79, 1, 2, 80, 81, 82, 83, 84, 85, 3, 15	cdleml6 40506	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝑈 ∈ 𝐸 ∧ (𝑈‘(𝑠‘ℎ)) = ℎ))
87	86	simpld 493	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → 𝑈 ∈ 𝐸)
88	75, 76, 77, 87	syl3anc 1368	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → 𝑈 ∈ 𝐸)
89	12	oveqd 7430	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑠𝑀𝑈) = (𝑠(._r‘𝐷)𝑈))
90	89	ad2antrr 724	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝑠𝑀𝑈) = (𝑠(._r‘𝐷)𝑈))
91		simprl 769	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → 𝑠 ∈ 𝐸)
92	1, 2, 3, 4, 10	erngmul-rN 40339	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝑈 ∈ 𝐸)) → (𝑠(._r‘𝐷)𝑈) = (𝑈 ∘ 𝑠))
93	75, 91, 88, 92	syl12anc 835	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝑠(._r‘𝐷)𝑈) = (𝑈 ∘ 𝑠))
94	14, 78, 79, 1, 2, 80, 81, 82, 83, 84, 85, 3, 15	cdleml8 40508	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝑈 ∘ 𝑠) = ( I ↾ 𝑇))
95	94	3expa 1115	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝑈 ∘ 𝑠) = ( I ↾ 𝑇))
96	93, 95	eqtrd 2765	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝑠(._r‘𝐷)𝑈) = ( I ↾ 𝑇))
97	90, 96	eqtrd 2765	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 𝑂)) → (𝑠𝑀𝑈) = ( I ↾ 𝑇))
98	8, 13, 33, 58, 59, 72, 74, 88, 97	isdrngrd 20657	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵))) → 𝐷 ∈ DivRing)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ifcif 4525 ↦ cmpt 5227 I cid 5570 ^◡ccnv 5672 ↾ cres 5675 ∘ ccom 5677 ‘cfv 6543 ℩crio 7368 (class class class)co 7413 ∈ cmpo 7415 Basecbs 17174 +_gcplusg 17227 ._rcmulr 17228 occoc 17235 0_gc0g 17415 joincjn 18297 meetcmee 18298 Grpcgrp 18889 1_rcur 20120 Ringcrg 20172 DivRingcdr 20623 HLchlt 38874 LHypclh 39509 LTrncltrn 39626 trLctrl 39683 TEndoctendo 40277 EDRing_Rcedring-rN 40279

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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-riotaBAD 38477

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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-0g 17417 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18892 df-minusg 18893 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-drng 20625 df-oposet 38700 df-ol 38702 df-oml 38703 df-covers 38790 df-ats 38791 df-atl 38822 df-cvlat 38846 df-hlat 38875 df-llines 39023 df-lplanes 39024 df-lvols 39025 df-lines 39026 df-psubsp 39028 df-pmap 39029 df-padd 39321 df-lhyp 39513 df-laut 39514 df-ldil 39629 df-ltrn 39630 df-trl 39684 df-tendo 40280 df-edring-rN 40281

This theorem is referenced by: erngdv-rN 40526

W3C validator