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Theorem smadiadetlem4 22392

Description: Lemma 4 for smadiadet 22393. (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 20136	. . . . . . . . 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 22133	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
8	7	simpld 494	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin)
10	4, 9	jca 511	. . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin))
11	10	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin))
12		simprl 768	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
13		simprr 770	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
14	6	eleq2i 2824	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴))
15	14	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴))
16	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
17	16	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐴))
18		eqid 2731	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
19	5, 18	matecl 22148	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
20	12, 13, 17, 19	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
21	2, 18	mgpbas 20035	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝐺)
22	20, 21	eleqtrdi 2842	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝐺))
23	22	ralrimivva 3199	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ (Base‘𝐺))
24	23	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ (Base‘𝐺))
25		crngring 20140	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
26		marep01ma.0	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑅)
27	18, 26	ring0cl 20156	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
28	1, 25, 27	mp2b 10	. . . . . . . 8 ⊢ 0 ∈ (Base‘𝑅)
29	28, 21	eleqtri 2830	. . . . . . 7 ⊢ 0 ∈ (Base‘𝐺)
30	24, 29	jctir 520	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ (Base‘𝐺) ∧ 0 ∈ (Base‘𝐺)))
31		simpr 484	. . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → 𝐾 ∈ 𝑁)
32	31	adantr 480	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) → 𝐾 ∈ 𝑁)
33		simpr 484	. . . . . 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 2731	. . . . . . 7 ⊢ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} = {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}
36		marep01ma.1	. . . . . . . 8 ⊢ 1 = (1_r‘𝑅)
37	2, 36	ringidval 20078	. . . . . . 7 ⊢ 1 = (0_g‘𝐺)
38		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
39	34, 35, 37, 38	gsummatr01 22382	. . . . . 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 1381	. . . . 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 7428	. . . 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 5249	. . 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 7428	. 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 22391	. 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 2771	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 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {crab 3431 Vcvv 3473 ∖ cdif 3946 ifcif 4529 {csn 4629 ↦ cmpt 5232 ∘ ccom 5681 ‘cfv 6544 (class class class)co 7412 ∈ cmpo 7414 Fincfn 8942 Basecbs 17149 ._rcmulr 17203 0_gc0g 17390 Σ_g cgsu 17391 SymGrpcsymg 19276 pmSgncpsgn 19399 CMndccmn 19690 mulGrpcmgp 20029 1_rcur 20076 Ringcrg 20128 CRingccrg 20129 ℤRHomczrh 21269 Mat cmat 22128

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-addf 11192 ax-mulf 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-sup 9440 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-xnn0 12550 df-z 12564 df-dec 12683 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-word 14470 df-lsw 14518 df-concat 14526 df-s1 14551 df-substr 14596 df-pfx 14626 df-splice 14705 df-reverse 14714 df-s2 14804 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-efmnd 18787 df-grp 18859 df-minusg 18860 df-mulg 18988 df-subg 19040 df-ghm 19129 df-gim 19174 df-cntz 19223 df-oppg 19252 df-symg 19277 df-pmtr 19352 df-psgn 19401 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-sra 20931 df-rgmod 20932 df-cnfld 21146 df-zring 21219 df-zrh 21273 df-dsmm 21507 df-frlm 21522 df-mat 22129

This theorem is referenced by: smadiadet 22393

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