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Theorem smadiadetlem4 22170

Description: Lemma 4 for smadiadet 22171. (Contributed by AV, 31-Jan-2019.)

**Hypotheses**
Ref	Expression
marep01ma.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
marep01ma.b	⊢ 𝐵 = (Base‘𝐴)
marep01ma.r	⊢ 𝑅 ∈ CRing
marep01ma.0	⊢ 0 = (0_g‘𝑅)
marep01ma.1	⊢ 1 = (1_r‘𝑅)
smadiadetlem.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
smadiadetlem.g	⊢ 𝐺 = (mulGrp‘𝑅)
madetminlem.y	⊢ 𝑌 = (ℤRHom‘𝑅)
madetminlem.s	⊢ 𝑆 = (pmSgn‘𝑁)
madetminlem.t	⊢ · = (._r‘𝑅)
smadiadetlem.w	⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
smadiadetlem.z	⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))

**Assertion**
Ref	Expression
smadiadetlem4	⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σ_g (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = (𝑅 Σ_g (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))

Distinct variable groups: 𝑖,𝑗,𝑛,𝐵 𝑖,𝑞,𝐾,𝑗,𝑛 𝑖,𝑀,𝑗,𝑛 𝑖,𝑁,𝑗,𝑛 𝑃,𝑖,𝑗,𝑛,𝑞 𝑅,𝑖,𝑗,𝑛 1 ,𝑖,𝑗,𝑛 0 ,𝑖,𝑗,𝑛 𝑛,𝐺,𝑝 𝐵,𝑝 𝐾,𝑝 𝑀,𝑝 𝑁,𝑝 𝑃,𝑝 𝑅,𝑝,𝑖,𝑗 𝑞,𝑝 𝑛,𝑊,𝑝 𝐺,𝑝 𝑌,𝑝 𝑍,𝑝 𝑖,𝐺,𝑗 Allowed substitution hints: 𝐴(𝑖,𝑗,𝑛,𝑞,𝑝) 𝐵(𝑞) 𝑅(𝑞) 𝑆(𝑖,𝑗,𝑛,𝑞,𝑝) · (𝑖,𝑗,𝑛,𝑞,𝑝) 1 (𝑞,𝑝) 𝐺(𝑞) 𝑀(𝑞) 𝑁(𝑞) 𝑊(𝑖,𝑗,𝑞) 𝑌(𝑖,𝑗,𝑛,𝑞) 0 (𝑞,𝑝) 𝑍(𝑖,𝑗,𝑛,𝑞)

**Proof of Theorem smadiadetlem4**
Step	Hyp	Ref	Expression
1		marep01ma.r	. . . . . . . . 9 ⊢ 𝑅 ∈ CRing
2		smadiadetlem.g	. . . . . . . . . 10 ⊢ 𝐺 = (mulGrp‘𝑅)
3	2	crngmgp 20063	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd)
4	1, 3	mp1i 13	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → 𝐺 ∈ CMnd)
5		marep01ma.a	. . . . . . . . . . 11 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6		marep01ma.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐴)
7	5, 6	matrcl 21911	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
8	7	simpld 495	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
9	8	adantr 481	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin)
10	4, 9	jca 512	. . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin))
11	10	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin))
12		simprl 769	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
13		simprr 771	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
14	6	eleq2i 2825	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴))
15	14	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴))
16	15	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
17	16	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐴))
18		eqid 2732	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
19	5, 18	matecl 21926	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
20	12, 13, 17, 19	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
21	2, 18	mgpbas 19992	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝐺)
22	20, 21	eleqtrdi 2843	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝐺))
23	22	ralrimivva 3200	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ (Base‘𝐺))
24	23	adantr 481	. . . . . . 7 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ (Base‘𝐺))
25		crngring 20067	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
26		marep01ma.0	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑅)
27	18, 26	ring0cl 20083	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
28	1, 25, 27	mp2b 10	. . . . . . . 8 ⊢ 0 ∈ (Base‘𝑅)
29	28, 21	eleqtri 2831	. . . . . . 7 ⊢ 0 ∈ (Base‘𝐺)
30	24, 29	jctir 521	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)))
31		simpr 485	. . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → 𝐾 ∈ 𝑁)
32	31	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → 𝐾 ∈ 𝑁)
33		simpr 485	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾})
34		smadiadetlem.p	. . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
35		eqid 2732	. . . . . . 7 ⊢ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
36		marep01ma.1	. . . . . . . 8 ⊢ 1 = (1_r‘𝑅)
37	2, 36	ringidval 20005	. . . . . . 7 ⊢ 1 = (0_g‘𝐺)
38		eqid 2732	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
39	34, 35, 37, 38	gsummatr01 22160	. . . . . 6 ⊢ (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁 ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾})) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛)))) = (𝐺 Σ_g (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛)))))
40	11, 30, 32, 32, 33, 39	syl113anc 1382	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛)))) = (𝐺 Σ_g (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛)))))
41	40	oveq2d 7424	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (((𝑌 ∘ 𝑆)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))) = (((𝑌 ∘ 𝑆)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))
42	41	mpteq2dva 5248	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛)))))) = (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛)))))))
43	42	oveq2d 7424	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σ_g (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = (𝑅 Σ_g (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))
44		madetminlem.y	. . 3 ⊢ 𝑌 = (ℤRHom‘𝑅)
45		madetminlem.s	. . 3 ⊢ 𝑆 = (pmSgn‘𝑁)
46		madetminlem.t	. . 3 ⊢ · = (._r‘𝑅)
47		smadiadetlem.w	. . 3 ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
48		smadiadetlem.z	. . 3 ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))
49	5, 6, 1, 26, 36, 34, 2, 44, 45, 46, 47, 48	smadiadetlem3 22169	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σ_g (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))) = (𝑅 Σ_g (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))
50	43, 49	eqtrd 2772	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σ_g (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = (𝑅 Σ_g (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(._r‘𝑅)(𝐺 Σ_g (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 Vcvv 3474 ∖ cdif 3945 ifcif 4528 {csn 4628 ↦ cmpt 5231 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 Fincfn 8938 Basecbs 17143 ._rcmulr 17197 0_gc0g 17384 Σ_g cgsu 17385 SymGrpcsymg 19233 pmSgncpsgn 19356 CMndccmn 19647 mulGrpcmgp 19986 1_rcur 20003 Ringcrg 20055 CRingccrg 20056 ℤRHomczrh 21048 Mat cmat 21906

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-xnn0 12544 df-z 12558 df-dec 12677 df-uz 12822 df-rp 12974 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-hash 14290 df-word 14464 df-lsw 14512 df-concat 14520 df-s1 14545 df-substr 14590 df-pfx 14620 df-splice 14699 df-reverse 14708 df-s2 14798 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-efmnd 18749 df-grp 18821 df-minusg 18822 df-mulg 18950 df-subg 19002 df-ghm 19089 df-gim 19132 df-cntz 19180 df-oppg 19209 df-symg 19234 df-pmtr 19309 df-psgn 19358 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-rnghom 20250 df-subrg 20316 df-sra 20784 df-rgmod 20785 df-cnfld 20944 df-zring 21017 df-zrh 21052 df-dsmm 21286 df-frlm 21301 df-mat 21907

This theorem is referenced by: smadiadet 22171

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