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Theorem mnringmulrcld 44223
Description: Monoid rings are closed under multiplication. (Contributed by Rohan Ridenour, 14-May-2024.)
Hypotheses
Ref Expression
mnringmulrcld.2 𝐹 = (𝑅 MndRing 𝑀)
mnringmulrcld.3 𝐵 = (Base‘𝐹)
mnringmulrcld.1 𝐴 = (Base‘𝑀)
mnringmulrcld.4 · = (.r𝐹)
mnringmulrcld.5 (𝜑𝑅 ∈ Ring)
mnringmulrcld.6 (𝜑𝑀𝑈)
mnringmulrcld.7 (𝜑𝑋𝐵)
mnringmulrcld.8 (𝜑𝑌𝐵)
Assertion
Ref Expression
mnringmulrcld (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)

Proof of Theorem mnringmulrcld
Dummy variables 𝑎 𝑏 𝑝 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnringmulrcld.2 . . 3 𝐹 = (𝑅 MndRing 𝑀)
2 mnringmulrcld.3 . . 3 𝐵 = (Base‘𝐹)
3 eqid 2734 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2734 . . 3 (0g𝑅) = (0g𝑅)
5 mnringmulrcld.1 . . 3 𝐴 = (Base‘𝑀)
6 eqid 2734 . . 3 (+g𝑀) = (+g𝑀)
7 mnringmulrcld.4 . . 3 · = (.r𝐹)
8 mnringmulrcld.5 . . 3 (𝜑𝑅 ∈ Ring)
9 mnringmulrcld.6 . . 3 (𝜑𝑀𝑈)
10 mnringmulrcld.7 . . 3 (𝜑𝑋𝐵)
11 mnringmulrcld.8 . . 3 (𝜑𝑌𝐵)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11mnringmulrvald 44222 . 2 (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))))
13 eqid 2734 . . 3 (0g𝐹) = (0g𝐹)
141, 8, 9mnringlmodd 44221 . . . 4 (𝜑𝐹 ∈ LMod)
15 lmodcmn 20924 . . . 4 (𝐹 ∈ LMod → 𝐹 ∈ CMnd)
1614, 15syl 17 . . 3 (𝜑𝐹 ∈ CMnd)
175fvexi 6920 . . . . 5 𝐴 ∈ V
1817, 17xpex 7771 . . . 4 (𝐴 × 𝐴) ∈ V
1918a1i 11 . . 3 (𝜑 → (𝐴 × 𝐴) ∈ V)
2083ad2ant1 1132 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → 𝑅 ∈ Ring)
21 eqid 2734 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘𝑅)
221, 2, 5, 21, 8, 9, 10mnringbasefd 44210 . . . . . . . . . . . . . . 15 (𝜑𝑋:𝐴⟶(Base‘𝑅))
23223ad2ant1 1132 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑋:𝐴⟶(Base‘𝑅))
24 simp2 1136 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑎𝐴)
2523, 24ffvelcdmd 7104 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (𝑋𝑎) ∈ (Base‘𝑅))
261, 2, 5, 21, 8, 9, 11mnringbasefd 44210 . . . . . . . . . . . . . . 15 (𝜑𝑌:𝐴⟶(Base‘𝑅))
27263ad2ant1 1132 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑌:𝐴⟶(Base‘𝑅))
28 simp3 1137 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑏𝐴)
2927, 28ffvelcdmd 7104 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (𝑌𝑏) ∈ (Base‘𝑅))
3021, 3ringcl 20267 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑋𝑎) ∈ (Base‘𝑅) ∧ (𝑌𝑏) ∈ (Base‘𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) ∈ (Base‘𝑅))
3120, 25, 29, 30syl3anc 1370 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) ∈ (Base‘𝑅))
3221, 4ring0cl 20280 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
3320, 32syl 17 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → (0g𝑅) ∈ (Base‘𝑅))
3431, 33ifcld 4576 . . . . . . . . . . 11 ((𝜑𝑎𝐴𝑏𝐴) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) ∈ (Base‘𝑅))
3534adantr 480 . . . . . . . . . 10 (((𝜑𝑎𝐴𝑏𝐴) ∧ 𝑖𝐴) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) ∈ (Base‘𝑅))
3635fmpttd 7134 . . . . . . . . 9 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))):𝐴⟶(Base‘𝑅))
3721fvexi 6920 . . . . . . . . . 10 (Base‘𝑅) ∈ V
3837, 17elmap 8909 . . . . . . . . 9 ((𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴) ↔ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))):𝐴⟶(Base‘𝑅))
3936, 38sylibr 234 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴))
4017a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐴𝑏𝐴) → 𝐴 ∈ V)
41 eqid 2734 . . . . . . . . 9 (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))
4240, 33, 41sniffsupp 9437 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) finSupp (0g𝑅))
4339, 42jca 511 . . . . . . 7 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴) ∧ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) finSupp (0g𝑅)))
4493ad2ant1 1132 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → 𝑀𝑈)
451, 2, 5, 21, 4, 20, 44mnringelbased 44209 . . . . . . 7 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵 ↔ ((𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴) ∧ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) finSupp (0g𝑅))))
4643, 45mpbird 257 . . . . . 6 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵)
47463expb 1119 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵)
4847ralrimivva 3199 . . . 4 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵)
49 eqid 2734 . . . . 5 (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) = (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
5049fmpo 8091 . . . 4 (∀𝑎𝐴𝑏𝐴 (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵 ↔ (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))):(𝐴 × 𝐴)⟶𝐵)
5148, 50sylib 218 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))):(𝐴 × 𝐴)⟶𝐵)
5217, 17mpoex 8102 . . . . 5 (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) ∈ V
5352a1i 11 . . . 4 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) ∈ V)
5451ffnd 6737 . . . 4 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) Fn (𝐴 × 𝐴))
5513fvexi 6920 . . . . 5 (0g𝐹) ∈ V
5655a1i 11 . . . 4 (𝜑 → (0g𝐹) ∈ V)
571, 2, 4, 8, 9, 10mnringbasefsuppd 44211 . . . . . 6 (𝜑𝑋 finSupp (0g𝑅))
5857fsuppimpd 9406 . . . . 5 (𝜑 → (𝑋 supp (0g𝑅)) ∈ Fin)
591, 2, 4, 8, 9, 11mnringbasefsuppd 44211 . . . . . 6 (𝜑𝑌 finSupp (0g𝑅))
6059fsuppimpd 9406 . . . . 5 (𝜑 → (𝑌 supp (0g𝑅)) ∈ Fin)
61 xpfi 9355 . . . . 5 (((𝑋 supp (0g𝑅)) ∈ Fin ∧ (𝑌 supp (0g𝑅)) ∈ Fin) → ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∈ Fin)
6258, 60, 61syl2anc 584 . . . 4 (𝜑 → ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∈ Fin)
63 elxpi 5710 . . . . . . 7 (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)))
64 simpl 482 . . . . . . . 8 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)) → 𝑝 = ⟨𝑎, 𝑏⟩)
65642eximi 1832 . . . . . . 7 (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)) → ∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
6663, 65syl 17 . . . . . 6 (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
6766adantl 481 . . . . 5 ((𝜑𝑝 ∈ (𝐴 × 𝐴)) → ∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
68 nfv 1911 . . . . . 6 𝑎(𝜑𝑝 ∈ (𝐴 × 𝐴))
69 nfv 1911 . . . . . . 7 𝑎 𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))
70 nfmpo1 7512 . . . . . . . . 9 𝑎(𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
71 nfcv 2902 . . . . . . . . 9 𝑎𝑝
7270, 71nffv 6916 . . . . . . . 8 𝑎((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝)
73 nfcv 2902 . . . . . . . 8 𝑎(0g𝐹)
7472, 73nfeq 2916 . . . . . . 7 𝑎((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)
7569, 74nfor 1901 . . . . . 6 𝑎(𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∨ ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹))
76 nfv 1911 . . . . . . 7 𝑏(𝜑𝑝 ∈ (𝐴 × 𝐴))
77 nfv 1911 . . . . . . . 8 𝑏 𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))
78 nfmpo2 7513 . . . . . . . . . 10 𝑏(𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
79 nfcv 2902 . . . . . . . . . 10 𝑏𝑝
8078, 79nffv 6916 . . . . . . . . 9 𝑏((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝)
81 nfcv 2902 . . . . . . . . 9 𝑏(0g𝐹)
8280, 81nfeq 2916 . . . . . . . 8 𝑏((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)
8377, 82nfor 1901 . . . . . . 7 𝑏(𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∨ ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹))
84 simp3 1137 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
85 simp2 1136 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 ∈ (𝐴 × 𝐴))
8684, 85eqeltrrd 2839 . . . . . . . . . 10 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴))
87 opelxp 5724 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴) ↔ (𝑎𝐴𝑏𝐴))
8886, 87sylib 218 . . . . . . . . 9 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑎𝐴𝑏𝐴))
89 ianor 983 . . . . . . . . . . . . . . . 16 (¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ↔ (¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g𝑅))))
9022ffnd 6737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 Fn 𝐴)
9117a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐴 ∈ V)
924fvexi 6920 . . . . . . . . . . . . . . . . . . . . . . . 24 (0g𝑅) ∈ V
9392a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (0g𝑅) ∈ V)
94 elsuppfn 8193 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 Fn 𝐴𝐴 ∈ V ∧ (0g𝑅) ∈ V) → (𝑎 ∈ (𝑋 supp (0g𝑅)) ↔ (𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅))))
9590, 91, 93, 94syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎 ∈ (𝑋 supp (0g𝑅)) ↔ (𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅))))
9695biimprd 248 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅)) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
97963ad2ant1 1132 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅)) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
9824, 97mpand 695 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎) ≠ (0g𝑅) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
9998necon1bd 2955 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝐴𝑏𝐴) → (¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) → (𝑋𝑎) = (0g𝑅)))
10026ffnd 6737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑌 Fn 𝐴)
101 elsuppfn 8193 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌 Fn 𝐴𝐴 ∈ V ∧ (0g𝑅) ∈ V) → (𝑏 ∈ (𝑌 supp (0g𝑅)) ↔ (𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅))))
102100, 91, 93, 101syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑏 ∈ (𝑌 supp (0g𝑅)) ↔ (𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅))))
103102biimprd 248 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅)) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
1041033ad2ant1 1132 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅)) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
10528, 104mpand 695 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑌𝑏) ≠ (0g𝑅) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
106105necon1bd 2955 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝐴𝑏𝐴) → (¬ 𝑏 ∈ (𝑌 supp (0g𝑅)) → (𝑌𝑏) = (0g𝑅)))
10799, 106orim12d 966 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝐴𝑏𝐴) → ((¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g𝑅))) → ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))))
108107imp 406 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ (¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g𝑅)))) → ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅)))
10989, 108sylan2b 594 . . . . . . . . . . . . . . 15 (((𝜑𝑎𝐴𝑏𝐴) ∧ ¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))) → ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅)))
110 oveq1 7437 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋𝑎) = (0g𝑅) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = ((0g𝑅)(.r𝑅)(𝑌𝑏)))
11121, 3, 4ringlz 20306 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝑌𝑏) ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
11220, 29, 111syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎𝐴𝑏𝐴) → ((0g𝑅)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
113110, 112sylan9eqr 2796 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎𝐴𝑏𝐴) ∧ (𝑋𝑎) = (0g𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
114 oveq2 7438 . . . . . . . . . . . . . . . . . . . . 21 ((𝑌𝑏) = (0g𝑅) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = ((𝑋𝑎)(.r𝑅)(0g𝑅)))
11521, 3, 4ringrz 20307 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝑋𝑎) ∈ (Base‘𝑅)) → ((𝑋𝑎)(.r𝑅)(0g𝑅)) = (0g𝑅))
11620, 25, 115syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎)(.r𝑅)(0g𝑅)) = (0g𝑅))
117114, 116sylan9eqr 2796 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎𝐴𝑏𝐴) ∧ (𝑌𝑏) = (0g𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
118113, 117jaodan 959 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
119118adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) ∧ 𝑖 = (𝑎(+g𝑀)𝑏)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
120 eqidd 2735 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) ∧ ¬ 𝑖 = (𝑎(+g𝑀)𝑏)) → (0g𝑅) = (0g𝑅))
121119, 120ifeqda 4566 . . . . . . . . . . . . . . . . 17 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) = (0g𝑅))
122121mpteq2dv 5249 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (𝑖𝐴 ↦ (0g𝑅)))
123 fconstmpt 5750 . . . . . . . . . . . . . . . . . . 19 (𝐴 × {(0g𝑅)}) = (𝑖𝐴 ↦ (0g𝑅))
1241, 4, 5, 8, 9mnring0g2d 44215 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝐹))
125123, 124eqtr3id 2788 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
1261253ad2ant1 1132 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
127126adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
128122, 127eqtrd 2774 . . . . . . . . . . . . . . 15 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹))
129109, 128syldan 591 . . . . . . . . . . . . . 14 (((𝜑𝑎𝐴𝑏𝐴) ∧ ¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹))
130129ex 412 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
131130orrd 863 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
1321313expb 1119 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
1331323adant3 1131 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
134 eleq1 2826 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ ⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))))
135 opelxp 5724 . . . . . . . . . . . . 13 (⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))))
136134, 135bitrdi 287 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))))
1371363ad2ant3 1134 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))))
138 simp2l 1198 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑎𝐴)
139 simp2r 1199 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑏𝐴)
140 eqidd 2735 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) = (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))))
141 simp3 1137 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
14217mptex 7242 . . . . . . . . . . . . . . 15 (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ V
143142a1i 11 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) ∧ 𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ V)
144140, 141, 143fvmpopr2d 7594 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) ∧ 𝑎𝐴𝑏𝐴) → ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
145138, 139, 144mpd3an23 1462 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
146145eqeq1d 2736 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹) ↔ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
147137, 146orbi12d 918 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ((𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∨ ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)) ↔ ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹))))
148133, 147mpbird 257 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∨ ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)))
14988, 148syld3an2 1410 . . . . . . . 8 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∨ ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)))
1501493expia 1120 . . . . . . 7 ((𝜑𝑝 ∈ (𝐴 × 𝐴)) → (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∨ ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹))))
15176, 83, 150exlimd 2215 . . . . . 6 ((𝜑𝑝 ∈ (𝐴 × 𝐴)) → (∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∨ ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹))))
15268, 75, 151exlimd 2215 . . . . 5 ((𝜑𝑝 ∈ (𝐴 × 𝐴)) → (∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∨ ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹))))
15367, 152mpd 15 . . . 4 ((𝜑𝑝 ∈ (𝐴 × 𝐴)) → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∨ ((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)))
15453, 54, 56, 62, 153finnzfsuppd 9410 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) finSupp (0g𝐹))
1552, 13, 16, 19, 51, 154gsumcl 19947 . 2 (𝜑 → (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))) ∈ 𝐵)
15612, 155eqeltrd 2838 1 (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  Vcvv 3477  ifcif 4530  {csn 4630  cop 4636   class class class wbr 5147  cmpt 5230   × cxp 5686   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432   supp csupp 8183  m cmap 8864  Fincfn 8983   finSupp cfsupp 9398  Basecbs 17244  +gcplusg 17297  .rcmulr 17298  0gc0g 17485   Σg cgsu 17486  CMndccmn 19812  Ringcrg 20250  LModclmod 20874   MndRing cmnring 44201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-sbg 18968  df-subg 19153  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-subrg 20586  df-lmod 20876  df-lss 20947  df-sra 21189  df-rgmod 21190  df-dsmm 21769  df-frlm 21784  df-mnring 44202
This theorem is referenced by: (None)
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