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Theorem mnringmulrcld 44247
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 2737 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2737 . . 3 (0g𝑅) = (0g𝑅)
5 mnringmulrcld.1 . . 3 𝐴 = (Base‘𝑀)
6 eqid 2737 . . 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 44246 . 2 (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))))
13 eqid 2737 . . 3 (0g𝐹) = (0g𝐹)
141, 8, 9mnringlmodd 44245 . . . 4 (𝜑𝐹 ∈ LMod)
15 lmodcmn 20908 . . . 4 (𝐹 ∈ LMod → 𝐹 ∈ CMnd)
1614, 15syl 17 . . 3 (𝜑𝐹 ∈ CMnd)
175fvexi 6920 . . . . 5 𝐴 ∈ V
1817, 17xpex 7773 . . . 4 (𝐴 × 𝐴) ∈ V
1918a1i 11 . . 3 (𝜑 → (𝐴 × 𝐴) ∈ V)
2083ad2ant1 1134 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → 𝑅 ∈ Ring)
21 eqid 2737 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘𝑅)
221, 2, 5, 21, 8, 9, 10mnringbasefd 44234 . . . . . . . . . . . . . . 15 (𝜑𝑋:𝐴⟶(Base‘𝑅))
23223ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑋:𝐴⟶(Base‘𝑅))
24 simp2 1138 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑎𝐴)
2523, 24ffvelcdmd 7105 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (𝑋𝑎) ∈ (Base‘𝑅))
261, 2, 5, 21, 8, 9, 11mnringbasefd 44234 . . . . . . . . . . . . . . 15 (𝜑𝑌:𝐴⟶(Base‘𝑅))
27263ad2ant1 1134 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑌:𝐴⟶(Base‘𝑅))
28 simp3 1139 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑏𝐴)
2927, 28ffvelcdmd 7105 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (𝑌𝑏) ∈ (Base‘𝑅))
3021, 3ringcl 20247 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑋𝑎) ∈ (Base‘𝑅) ∧ (𝑌𝑏) ∈ (Base‘𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) ∈ (Base‘𝑅))
3120, 25, 29, 30syl3anc 1373 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) ∈ (Base‘𝑅))
3221, 4ring0cl 20264 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
3320, 32syl 17 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → (0g𝑅) ∈ (Base‘𝑅))
3431, 33ifcld 4572 . . . . . . . . . . 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 7135 . . . . . . . . 9 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))):𝐴⟶(Base‘𝑅))
3721fvexi 6920 . . . . . . . . . 10 (Base‘𝑅) ∈ V
3837, 17elmap 8911 . . . . . . . . 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 2737 . . . . . . . . 9 (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))
4240, 33, 41sniffsupp 9440 . . . . . . . 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 1134 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → 𝑀𝑈)
451, 2, 5, 21, 4, 20, 44mnringelbased 44233 . . . . . . 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 1121 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵)
4847ralrimivva 3202 . . . 4 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵)
49 eqid 2737 . . . . 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 8093 . . . 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 8104 . . . . 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 44235 . . . . . 6 (𝜑𝑋 finSupp (0g𝑅))
5857fsuppimpd 9409 . . . . 5 (𝜑 → (𝑋 supp (0g𝑅)) ∈ Fin)
591, 2, 4, 8, 9, 11mnringbasefsuppd 44235 . . . . . 6 (𝜑𝑌 finSupp (0g𝑅))
6059fsuppimpd 9409 . . . . 5 (𝜑 → (𝑌 supp (0g𝑅)) ∈ Fin)
61 xpfi 9358 . . . . 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 5707 . . . . . . 7 (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)))
64 simpl 482 . . . . . . . 8 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)) → 𝑝 = ⟨𝑎, 𝑏⟩)
65642eximi 1836 . . . . . . 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 1914 . . . . . 6 𝑎(𝜑𝑝 ∈ (𝐴 × 𝐴))
69 nfv 1914 . . . . . . 7 𝑎 𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))
70 nfmpo1 7513 . . . . . . . . 9 𝑎(𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
71 nfcv 2905 . . . . . . . . 9 𝑎𝑝
7270, 71nffv 6916 . . . . . . . 8 𝑎((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝)
73 nfcv 2905 . . . . . . . 8 𝑎(0g𝐹)
7472, 73nfeq 2919 . . . . . . 7 𝑎((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)
7569, 74nfor 1904 . . . . . 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 1914 . . . . . . 7 𝑏(𝜑𝑝 ∈ (𝐴 × 𝐴))
77 nfv 1914 . . . . . . . 8 𝑏 𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))
78 nfmpo2 7514 . . . . . . . . . 10 𝑏(𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
79 nfcv 2905 . . . . . . . . . 10 𝑏𝑝
8078, 79nffv 6916 . . . . . . . . 9 𝑏((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝)
81 nfcv 2905 . . . . . . . . 9 𝑏(0g𝐹)
8280, 81nfeq 2919 . . . . . . . 8 𝑏((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)
8377, 82nfor 1904 . . . . . . 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 1139 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
85 simp2 1138 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 ∈ (𝐴 × 𝐴))
8684, 85eqeltrrd 2842 . . . . . . . . . 10 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴))
87 opelxp 5721 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴) ↔ (𝑎𝐴𝑏𝐴))
8886, 87sylib 218 . . . . . . . . 9 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑎𝐴𝑏𝐴))
89 ianor 984 . . . . . . . . . . . . . . . 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 8195 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 Fn 𝐴𝐴 ∈ V ∧ (0g𝑅) ∈ V) → (𝑎 ∈ (𝑋 supp (0g𝑅)) ↔ (𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅))))
9590, 91, 93, 94syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎 ∈ (𝑋 supp (0g𝑅)) ↔ (𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅))))
9695biimprd 248 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅)) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
97963ad2ant1 1134 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅)) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
9824, 97mpand 695 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎) ≠ (0g𝑅) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
9998necon1bd 2958 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝐴𝑏𝐴) → (¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) → (𝑋𝑎) = (0g𝑅)))
10026ffnd 6737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑌 Fn 𝐴)
101 elsuppfn 8195 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌 Fn 𝐴𝐴 ∈ V ∧ (0g𝑅) ∈ V) → (𝑏 ∈ (𝑌 supp (0g𝑅)) ↔ (𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅))))
102100, 91, 93, 101syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑏 ∈ (𝑌 supp (0g𝑅)) ↔ (𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅))))
103102biimprd 248 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅)) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
1041033ad2ant1 1134 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅)) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
10528, 104mpand 695 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑌𝑏) ≠ (0g𝑅) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
106105necon1bd 2958 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝐴𝑏𝐴) → (¬ 𝑏 ∈ (𝑌 supp (0g𝑅)) → (𝑌𝑏) = (0g𝑅)))
10799, 106orim12d 967 . . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋𝑎) = (0g𝑅) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = ((0g𝑅)(.r𝑅)(𝑌𝑏)))
11121, 3, 4ringlz 20290 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝑌𝑏) ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
11220, 29, 111syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎𝐴𝑏𝐴) → ((0g𝑅)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
113110, 112sylan9eqr 2799 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎𝐴𝑏𝐴) ∧ (𝑋𝑎) = (0g𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
114 oveq2 7439 . . . . . . . . . . . . . . . . . . . . 21 ((𝑌𝑏) = (0g𝑅) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = ((𝑋𝑎)(.r𝑅)(0g𝑅)))
11521, 3, 4ringrz 20291 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝑋𝑎) ∈ (Base‘𝑅)) → ((𝑋𝑎)(.r𝑅)(0g𝑅)) = (0g𝑅))
11620, 25, 115syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎)(.r𝑅)(0g𝑅)) = (0g𝑅))
117114, 116sylan9eqr 2799 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎𝐴𝑏𝐴) ∧ (𝑌𝑏) = (0g𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
118113, 117jaodan 960 . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) ∧ ¬ 𝑖 = (𝑎(+g𝑀)𝑏)) → (0g𝑅) = (0g𝑅))
121119, 120ifeqda 4562 . . . . . . . . . . . . . . . . 17 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) = (0g𝑅))
122121mpteq2dv 5244 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (𝑖𝐴 ↦ (0g𝑅)))
123 fconstmpt 5747 . . . . . . . . . . . . . . . . . . 19 (𝐴 × {(0g𝑅)}) = (𝑖𝐴 ↦ (0g𝑅))
1241, 4, 5, 8, 9mnring0g2d 44239 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝐹))
125123, 124eqtr3id 2791 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
1261253ad2ant1 1134 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
127126adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
128122, 127eqtrd 2777 . . . . . . . . . . . . . . 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 864 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
1321313expb 1121 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
1331323adant3 1133 . . . . . . . . . 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 2829 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ ⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))))
135 opelxp 5721 . . . . . . . . . . . . 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 1136 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))))
138 simp2l 1200 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑎𝐴)
139 simp2r 1201 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑏𝐴)
140 eqidd 2738 . . . . . . . . . . . . . 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 1139 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
14217mptex 7243 . . . . . . . . . . . . . . 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 7595 . . . . . . . . . . . . 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 1465 . . . . . . . . . . . 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 2739 . . . . . . . . . . 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 919 . . . . . . . . . 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 1413 . . . . . . . 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 1122 . . . . . . 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 2218 . . . . . 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 2218 . . . . 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 9413 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) finSupp (0g𝐹))
1552, 13, 16, 19, 51, 154gsumcl 19933 . 2 (𝜑 → (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))) ∈ 𝐵)
15612, 155eqeltrd 2841 1 (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  Vcvv 3480  ifcif 4525  {csn 4626  cop 4632   class class class wbr 5143  cmpt 5225   × cxp 5683   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433   supp csupp 8185  m cmap 8866  Fincfn 8985   finSupp cfsupp 9401  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  0gc0g 17484   Σg cgsu 17485  CMndccmn 19798  Ringcrg 20230  LModclmod 20858   MndRing cmnring 44225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  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 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-dsmm 21752  df-frlm 21767  df-mnring 44226
This theorem is referenced by: (None)
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