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Theorem mnringmulrcld 40870
 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 2822 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2822 . . 3 (0g𝑅) = (0g𝑅)
5 mnringmulrcld.1 . . 3 𝐴 = (Base‘𝑀)
6 eqid 2822 . . 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 40869 . 2 (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))))
13 eqid 2822 . . 3 (0g𝐹) = (0g𝐹)
141, 8, 9mnringlmodd 40868 . . . 4 (𝜑𝐹 ∈ LMod)
15 lmodcmn 19673 . . . 4 (𝐹 ∈ LMod → 𝐹 ∈ CMnd)
1614, 15syl 17 . . 3 (𝜑𝐹 ∈ CMnd)
175fvexi 6666 . . . . 5 𝐴 ∈ V
1817, 17xpex 7461 . . . 4 (𝐴 × 𝐴) ∈ V
1918a1i 11 . . 3 (𝜑 → (𝐴 × 𝐴) ∈ V)
2083ad2ant1 1130 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → 𝑅 ∈ Ring)
21 eqid 2822 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘𝑅)
221, 2, 5, 21, 8, 9, 10mnringbasefd 40860 . . . . . . . . . . . . . . 15 (𝜑𝑋:𝐴⟶(Base‘𝑅))
23223ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑋:𝐴⟶(Base‘𝑅))
24 simp2 1134 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑎𝐴)
2523, 24ffvelrnd 6834 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (𝑋𝑎) ∈ (Base‘𝑅))
261, 2, 5, 21, 8, 9, 11mnringbasefd 40860 . . . . . . . . . . . . . . 15 (𝜑𝑌:𝐴⟶(Base‘𝑅))
27263ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑌:𝐴⟶(Base‘𝑅))
28 simp3 1135 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑏𝐴)
2927, 28ffvelrnd 6834 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (𝑌𝑏) ∈ (Base‘𝑅))
3021, 3ringcl 19305 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑋𝑎) ∈ (Base‘𝑅) ∧ (𝑌𝑏) ∈ (Base‘𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) ∈ (Base‘𝑅))
3120, 25, 29, 30syl3anc 1368 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) ∈ (Base‘𝑅))
3221, 4ring0cl 19313 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
3320, 32syl 17 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → (0g𝑅) ∈ (Base‘𝑅))
3431, 33ifcld 4484 . . . . . . . . . . 11 ((𝜑𝑎𝐴𝑏𝐴) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) ∈ (Base‘𝑅))
3534adantr 484 . . . . . . . . . 10 (((𝜑𝑎𝐴𝑏𝐴) ∧ 𝑖𝐴) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) ∈ (Base‘𝑅))
3635fmpttd 6861 . . . . . . . . 9 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))):𝐴⟶(Base‘𝑅))
3721fvexi 6666 . . . . . . . . . 10 (Base‘𝑅) ∈ V
3837, 17elmap 8422 . . . . . . . . 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 237 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴))
4017a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐴𝑏𝐴) → 𝐴 ∈ V)
41 eqid 2822 . . . . . . . . 9 (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))
4240, 33, 41sniffsupp 8861 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) finSupp (0g𝑅))
4339, 42jca 515 . . . . . . 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 1130 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → 𝑀𝑈)
451, 2, 5, 21, 4, 20, 44mnringelbased 40859 . . . . . . 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 260 . . . . . 6 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵)
47463expb 1117 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵)
4847ralrimivva 3181 . . . 4 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵)
49 eqid 2822 . . . . 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 7752 . . . 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 221 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))):(𝐴 × 𝐴)⟶𝐵)
5217, 17mpoex 7764 . . . . 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 6495 . . . 4 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) Fn (𝐴 × 𝐴))
5513fvexi 6666 . . . . 5 (0g𝐹) ∈ V
5655a1i 11 . . . 4 (𝜑 → (0g𝐹) ∈ V)
571, 2, 4, 8, 9, 10mnringbasefsuppd 40861 . . . . . 6 (𝜑𝑋 finSupp (0g𝑅))
5857fsuppimpd 8828 . . . . 5 (𝜑 → (𝑋 supp (0g𝑅)) ∈ Fin)
591, 2, 4, 8, 9, 11mnringbasefsuppd 40861 . . . . . 6 (𝜑𝑌 finSupp (0g𝑅))
6059fsuppimpd 8828 . . . . 5 (𝜑 → (𝑌 supp (0g𝑅)) ∈ Fin)
61 xpfi 8777 . . . . 5 (((𝑋 supp (0g𝑅)) ∈ Fin ∧ (𝑌 supp (0g𝑅)) ∈ Fin) → ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∈ Fin)
6258, 60, 61syl2anc 587 . . . 4 (𝜑 → ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ∈ Fin)
63 elxpi 5554 . . . . . . 7 (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)))
64 simpl 486 . . . . . . . 8 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)) → 𝑝 = ⟨𝑎, 𝑏⟩)
65642eximi 1837 . . . . . . 7 (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)) → ∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
6663, 65syl 17 . . . . . 6 (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
6766adantl 485 . . . . 5 ((𝜑𝑝 ∈ (𝐴 × 𝐴)) → ∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
68 nfv 1915 . . . . . 6 𝑎(𝜑𝑝 ∈ (𝐴 × 𝐴))
69 nfv 1915 . . . . . . 7 𝑎 𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))
70 nfmpo1 7218 . . . . . . . . 9 𝑎(𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
71 nfcv 2979 . . . . . . . . 9 𝑎𝑝
7270, 71nffv 6662 . . . . . . . 8 𝑎((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝)
73 nfcv 2979 . . . . . . . 8 𝑎(0g𝐹)
7472, 73nfeq 2992 . . . . . . 7 𝑎((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)
7569, 74nfor 1905 . . . . . 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 1915 . . . . . . 7 𝑏(𝜑𝑝 ∈ (𝐴 × 𝐴))
77 nfv 1915 . . . . . . . 8 𝑏 𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))
78 nfmpo2 7219 . . . . . . . . . 10 𝑏(𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
79 nfcv 2979 . . . . . . . . . 10 𝑏𝑝
8078, 79nffv 6662 . . . . . . . . 9 𝑏((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝)
81 nfcv 2979 . . . . . . . . 9 𝑏(0g𝐹)
8280, 81nfeq 2992 . . . . . . . 8 𝑏((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)
8377, 82nfor 1905 . . . . . . 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 1135 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
85 simp2 1134 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 ∈ (𝐴 × 𝐴))
8684, 85eqeltrrd 2915 . . . . . . . . . 10 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴))
87 opelxp 5568 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴) ↔ (𝑎𝐴𝑏𝐴))
8886, 87sylib 221 . . . . . . . . 9 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑎𝐴𝑏𝐴))
89 ianor 979 . . . . . . . . . . . . . . . 16 (¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ↔ (¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g𝑅))))
9022ffnd 6495 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 Fn 𝐴)
9117a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐴 ∈ V)
924fvexi 6666 . . . . . . . . . . . . . . . . . . . . . . . 24 (0g𝑅) ∈ V
9392a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (0g𝑅) ∈ V)
94 elsuppfn 7825 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 Fn 𝐴𝐴 ∈ V ∧ (0g𝑅) ∈ V) → (𝑎 ∈ (𝑋 supp (0g𝑅)) ↔ (𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅))))
9590, 91, 93, 94syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎 ∈ (𝑋 supp (0g𝑅)) ↔ (𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅))))
9695biimprd 251 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅)) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
97963ad2ant1 1130 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅)) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
9824, 97mpand 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎) ≠ (0g𝑅) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
9998necon1bd 3029 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝐴𝑏𝐴) → (¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) → (𝑋𝑎) = (0g𝑅)))
10026ffnd 6495 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑌 Fn 𝐴)
101 elsuppfn 7825 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌 Fn 𝐴𝐴 ∈ V ∧ (0g𝑅) ∈ V) → (𝑏 ∈ (𝑌 supp (0g𝑅)) ↔ (𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅))))
102100, 91, 93, 101syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑏 ∈ (𝑌 supp (0g𝑅)) ↔ (𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅))))
103102biimprd 251 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅)) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
1041033ad2ant1 1130 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅)) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
10528, 104mpand 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑌𝑏) ≠ (0g𝑅) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
106105necon1bd 3029 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝐴𝑏𝐴) → (¬ 𝑏 ∈ (𝑌 supp (0g𝑅)) → (𝑌𝑏) = (0g𝑅)))
10799, 106orim12d 962 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝐴𝑏𝐴) → ((¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g𝑅))) → ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))))
108107imp 410 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ (¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g𝑅)))) → ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅)))
10989, 108sylan2b 596 . . . . . . . . . . . . . . 15 (((𝜑𝑎𝐴𝑏𝐴) ∧ ¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))) → ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅)))
110 oveq1 7147 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋𝑎) = (0g𝑅) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = ((0g𝑅)(.r𝑅)(𝑌𝑏)))
11121, 3, 4ringlz 19331 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝑌𝑏) ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
11220, 29, 111syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎𝐴𝑏𝐴) → ((0g𝑅)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
113110, 112sylan9eqr 2879 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎𝐴𝑏𝐴) ∧ (𝑋𝑎) = (0g𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
114 oveq2 7148 . . . . . . . . . . . . . . . . . . . . 21 ((𝑌𝑏) = (0g𝑅) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = ((𝑋𝑎)(.r𝑅)(0g𝑅)))
11521, 3, 4ringrz 19332 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝑋𝑎) ∈ (Base‘𝑅)) → ((𝑋𝑎)(.r𝑅)(0g𝑅)) = (0g𝑅))
11620, 25, 115syl2anc 587 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎)(.r𝑅)(0g𝑅)) = (0g𝑅))
117114, 116sylan9eqr 2879 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎𝐴𝑏𝐴) ∧ (𝑌𝑏) = (0g𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
118113, 117jaodan 955 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
119118adantr 484 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) ∧ 𝑖 = (𝑎(+g𝑀)𝑏)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
120 eqidd 2823 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) ∧ ¬ 𝑖 = (𝑎(+g𝑀)𝑏)) → (0g𝑅) = (0g𝑅))
121119, 120ifeqda 4474 . . . . . . . . . . . . . . . . 17 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) = (0g𝑅))
122121mpteq2dv 5138 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (𝑖𝐴 ↦ (0g𝑅)))
123 fconstmpt 5591 . . . . . . . . . . . . . . . . . . 19 (𝐴 × {(0g𝑅)}) = (𝑖𝐴 ↦ (0g𝑅))
1241, 4, 5, 8, 9mnring0g2d 40864 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝐹))
125123, 124syl5eqr 2871 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
1261253ad2ant1 1130 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
127126adantr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
128122, 127eqtrd 2857 . . . . . . . . . . . . . . 15 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹))
129109, 128syldan 594 . . . . . . . . . . . . . 14 (((𝜑𝑎𝐴𝑏𝐴) ∧ ¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹))
130129ex 416 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
131130orrd 860 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
1321313expb 1117 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
1331323adant3 1129 . . . . . . . . . 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 2901 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ ⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))))
135 opelxp 5568 . . . . . . . . . . . . 13 (⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))))
136134, 135syl6bb 290 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))))
1371363ad2ant3 1132 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))))
138 simp2l 1196 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑎𝐴)
139 simp2r 1197 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑏𝐴)
140 eqidd 2823 . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
14217mptex 6968 . . . . . . . . . . . . . . 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 7295 . . . . . . . . . . . . 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 1460 . . . . . . . . . . . 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 2824 . . . . . . . . . . 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 916 . . . . . . . . . 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 260 . . . . . . . . 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 1408 . . . . . . . 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 1118 . . . . . . 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 2219 . . . . . 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 2219 . . . . 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 40849 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) finSupp (0g𝐹))
1552, 13, 16, 19, 51, 154gsumcl 19026 . 2 (𝜑 → (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))) ∈ 𝐵)
15612, 155eqeltrd 2914 1 (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2114   ≠ wne 3011  ∀wral 3130  Vcvv 3469  ifcif 4439  {csn 4539  ⟨cop 4545   class class class wbr 5042   ↦ cmpt 5122   × cxp 5530   Fn wfn 6329  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140   ∈ cmpo 7142   supp csupp 7817   ↑m cmap 8393  Fincfn 8496   finSupp cfsupp 8821  Basecbs 16474  +gcplusg 16556  .rcmulr 16557  0gc0g 16704   Σg cgsu 16705  CMndccmn 18897  Ringcrg 19288  LModclmod 19625   MndRing cmnring 40853 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-mulr 16570  df-sca 16572  df-vsca 16573  df-ip 16574  df-tset 16575  df-ple 16576  df-ds 16578  df-hom 16580  df-cco 16581  df-0g 16706  df-gsum 16707  df-prds 16712  df-pws 16714  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-minusg 18098  df-sbg 18099  df-subg 18267  df-cntz 18438  df-cmn 18899  df-abl 18900  df-mgp 19231  df-ur 19243  df-ring 19290  df-subrg 19524  df-lmod 19627  df-lss 19695  df-sra 19935  df-rgmod 19936  df-dsmm 20419  df-frlm 20434  df-mnring 40854 This theorem is referenced by: (None)
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