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Theorem mnringmulrcld 42756
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 2731 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2731 . . 3 (0g𝑅) = (0g𝑅)
5 mnringmulrcld.1 . . 3 𝐴 = (Base‘𝑀)
6 eqid 2731 . . 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 42755 . 2 (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))))
13 eqid 2731 . . 3 (0g𝐹) = (0g𝐹)
141, 8, 9mnringlmodd 42754 . . . 4 (𝜑𝐹 ∈ LMod)
15 lmodcmn 20469 . . . 4 (𝐹 ∈ LMod → 𝐹 ∈ CMnd)
1614, 15syl 17 . . 3 (𝜑𝐹 ∈ CMnd)
175fvexi 6892 . . . . 5 𝐴 ∈ V
1817, 17xpex 7723 . . . 4 (𝐴 × 𝐴) ∈ V
1918a1i 11 . . 3 (𝜑 → (𝐴 × 𝐴) ∈ V)
2083ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → 𝑅 ∈ Ring)
21 eqid 2731 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘𝑅)
221, 2, 5, 21, 8, 9, 10mnringbasefd 42743 . . . . . . . . . . . . . . 15 (𝜑𝑋:𝐴⟶(Base‘𝑅))
23223ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑋:𝐴⟶(Base‘𝑅))
24 simp2 1137 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑎𝐴)
2523, 24ffvelcdmd 7072 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (𝑋𝑎) ∈ (Base‘𝑅))
261, 2, 5, 21, 8, 9, 11mnringbasefd 42743 . . . . . . . . . . . . . . 15 (𝜑𝑌:𝐴⟶(Base‘𝑅))
27263ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑌:𝐴⟶(Base‘𝑅))
28 simp3 1138 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐴𝑏𝐴) → 𝑏𝐴)
2927, 28ffvelcdmd 7072 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (𝑌𝑏) ∈ (Base‘𝑅))
3021, 3ringcl 20031 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑋𝑎) ∈ (Base‘𝑅) ∧ (𝑌𝑏) ∈ (Base‘𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) ∈ (Base‘𝑅))
3120, 25, 29, 30syl3anc 1371 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) ∈ (Base‘𝑅))
3221, 4ring0cl 20041 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
3320, 32syl 17 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → (0g𝑅) ∈ (Base‘𝑅))
3431, 33ifcld 4568 . . . . . . . . . . 11 ((𝜑𝑎𝐴𝑏𝐴) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) ∈ (Base‘𝑅))
3534adantr 481 . . . . . . . . . 10 (((𝜑𝑎𝐴𝑏𝐴) ∧ 𝑖𝐴) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) ∈ (Base‘𝑅))
3635fmpttd 7099 . . . . . . . . 9 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))):𝐴⟶(Base‘𝑅))
3721fvexi 6892 . . . . . . . . . 10 (Base‘𝑅) ∈ V
3837, 17elmap 8848 . . . . . . . . 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 233 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴))
4017a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐴𝑏𝐴) → 𝐴 ∈ V)
41 eqid 2731 . . . . . . . . 9 (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))
4240, 33, 41sniffsupp 9377 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) finSupp (0g𝑅))
4339, 42jca 512 . . . . . . 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 1133 . . . . . . . 8 ((𝜑𝑎𝐴𝑏𝐴) → 𝑀𝑈)
451, 2, 5, 21, 4, 20, 44mnringelbased 42742 . . . . . . 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 256 . . . . . 6 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) ∈ 𝐵)
47463expb 1120 . . . . 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 2731 . . . . 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 8036 . . . 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 217 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))):(𝐴 × 𝐴)⟶𝐵)
5217, 17mpoex 8048 . . . . 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 6705 . . . 4 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) Fn (𝐴 × 𝐴))
5513fvexi 6892 . . . . 5 (0g𝐹) ∈ V
5655a1i 11 . . . 4 (𝜑 → (0g𝐹) ∈ V)
571, 2, 4, 8, 9, 10mnringbasefsuppd 42744 . . . . . 6 (𝜑𝑋 finSupp (0g𝑅))
5857fsuppimpd 9352 . . . . 5 (𝜑 → (𝑋 supp (0g𝑅)) ∈ Fin)
591, 2, 4, 8, 9, 11mnringbasefsuppd 42744 . . . . . 6 (𝜑𝑌 finSupp (0g𝑅))
6059fsuppimpd 9352 . . . . 5 (𝜑 → (𝑌 supp (0g𝑅)) ∈ Fin)
61 xpfi 9300 . . . . 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 5691 . . . . . . 7 (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)))
64 simpl 483 . . . . . . . 8 ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)) → 𝑝 = ⟨𝑎, 𝑏⟩)
65642eximi 1838 . . . . . . 7 (∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎𝐴𝑏𝐴)) → ∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
6663, 65syl 17 . . . . . 6 (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
6766adantl 482 . . . . 5 ((𝜑𝑝 ∈ (𝐴 × 𝐴)) → ∃𝑎𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
68 nfv 1917 . . . . . 6 𝑎(𝜑𝑝 ∈ (𝐴 × 𝐴))
69 nfv 1917 . . . . . . 7 𝑎 𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))
70 nfmpo1 7473 . . . . . . . . 9 𝑎(𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
71 nfcv 2902 . . . . . . . . 9 𝑎𝑝
7270, 71nffv 6888 . . . . . . . 8 𝑎((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝)
73 nfcv 2902 . . . . . . . 8 𝑎(0g𝐹)
7472, 73nfeq 2915 . . . . . . 7 𝑎((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)
7569, 74nfor 1907 . . . . . 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 1917 . . . . . . 7 𝑏(𝜑𝑝 ∈ (𝐴 × 𝐴))
77 nfv 1917 . . . . . . . 8 𝑏 𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))
78 nfmpo2 7474 . . . . . . . . . 10 𝑏(𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))
79 nfcv 2902 . . . . . . . . . 10 𝑏𝑝
8078, 79nffv 6888 . . . . . . . . 9 𝑏((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝)
81 nfcv 2902 . . . . . . . . 9 𝑏(0g𝐹)
8280, 81nfeq 2915 . . . . . . . 8 𝑏((𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))‘𝑝) = (0g𝐹)
8377, 82nfor 1907 . . . . . . 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 1138 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
85 simp2 1137 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 ∈ (𝐴 × 𝐴))
8684, 85eqeltrrd 2833 . . . . . . . . . 10 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴))
87 opelxp 5705 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴) ↔ (𝑎𝐴𝑏𝐴))
8886, 87sylib 217 . . . . . . . . 9 ((𝜑𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑎𝐴𝑏𝐴))
89 ianor 980 . . . . . . . . . . . . . . . 16 (¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ↔ (¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g𝑅))))
9022ffnd 6705 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 Fn 𝐴)
9117a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐴 ∈ V)
924fvexi 6892 . . . . . . . . . . . . . . . . . . . . . . . 24 (0g𝑅) ∈ V
9392a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (0g𝑅) ∈ V)
94 elsuppfn 8138 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 Fn 𝐴𝐴 ∈ V ∧ (0g𝑅) ∈ V) → (𝑎 ∈ (𝑋 supp (0g𝑅)) ↔ (𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅))))
9590, 91, 93, 94syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎 ∈ (𝑋 supp (0g𝑅)) ↔ (𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅))))
9695biimprd 247 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅)) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
97963ad2ant1 1133 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑎𝐴 ∧ (𝑋𝑎) ≠ (0g𝑅)) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
9824, 97mpand 693 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎) ≠ (0g𝑅) → 𝑎 ∈ (𝑋 supp (0g𝑅))))
9998necon1bd 2957 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝐴𝑏𝐴) → (¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) → (𝑋𝑎) = (0g𝑅)))
10026ffnd 6705 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑌 Fn 𝐴)
101 elsuppfn 8138 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑌 Fn 𝐴𝐴 ∈ V ∧ (0g𝑅) ∈ V) → (𝑏 ∈ (𝑌 supp (0g𝑅)) ↔ (𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅))))
102100, 91, 93, 101syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑏 ∈ (𝑌 supp (0g𝑅)) ↔ (𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅))))
103102biimprd 247 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅)) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
1041033ad2ant1 1133 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑏𝐴 ∧ (𝑌𝑏) ≠ (0g𝑅)) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
10528, 104mpand 693 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑌𝑏) ≠ (0g𝑅) → 𝑏 ∈ (𝑌 supp (0g𝑅))))
106105necon1bd 2957 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎𝐴𝑏𝐴) → (¬ 𝑏 ∈ (𝑌 supp (0g𝑅)) → (𝑌𝑏) = (0g𝑅)))
10799, 106orim12d 963 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝐴𝑏𝐴) → ((¬ 𝑎 ∈ (𝑋 supp (0g𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g𝑅))) → ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))))
108107imp 407 . . . . . . . . . . . . . . . 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 7400 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋𝑎) = (0g𝑅) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = ((0g𝑅)(.r𝑅)(𝑌𝑏)))
11121, 3, 4ringlz 20064 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝑌𝑏) ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
11220, 29, 111syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎𝐴𝑏𝐴) → ((0g𝑅)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
113110, 112sylan9eqr 2793 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎𝐴𝑏𝐴) ∧ (𝑋𝑎) = (0g𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
114 oveq2 7401 . . . . . . . . . . . . . . . . . . . . 21 ((𝑌𝑏) = (0g𝑅) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = ((𝑋𝑎)(.r𝑅)(0g𝑅)))
11521, 3, 4ringrz 20065 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ Ring ∧ (𝑋𝑎) ∈ (Base‘𝑅)) → ((𝑋𝑎)(.r𝑅)(0g𝑅)) = (0g𝑅))
11620, 25, 115syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑋𝑎)(.r𝑅)(0g𝑅)) = (0g𝑅))
117114, 116sylan9eqr 2793 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎𝐴𝑏𝐴) ∧ (𝑌𝑏) = (0g𝑅)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
118113, 117jaodan 956 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
119118adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) ∧ 𝑖 = (𝑎(+g𝑀)𝑏)) → ((𝑋𝑎)(.r𝑅)(𝑌𝑏)) = (0g𝑅))
120 eqidd 2732 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) ∧ ¬ 𝑖 = (𝑎(+g𝑀)𝑏)) → (0g𝑅) = (0g𝑅))
121119, 120ifeqda 4558 . . . . . . . . . . . . . . . . 17 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)) = (0g𝑅))
122121mpteq2dv 5243 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (𝑖𝐴 ↦ (0g𝑅)))
123 fconstmpt 5730 . . . . . . . . . . . . . . . . . . 19 (𝐴 × {(0g𝑅)}) = (𝑖𝐴 ↦ (0g𝑅))
1241, 4, 5, 8, 9mnring0g2d 42748 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝐹))
125123, 124eqtr3id 2785 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
1261253ad2ant1 1133 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎𝐴𝑏𝐴) → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
127126adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴𝑏𝐴) ∧ ((𝑋𝑎) = (0g𝑅) ∨ (𝑌𝑏) = (0g𝑅))) → (𝑖𝐴 ↦ (0g𝑅)) = (0g𝐹))
128122, 127eqtrd 2771 . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . 13 ((𝜑𝑎𝐴𝑏𝐴) → (¬ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) → (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
131130orrd 861 . . . . . . . . . . . 12 ((𝜑𝑎𝐴𝑏𝐴) → ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
1321313expb 1120 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐴)) → ((𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))) ∨ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))) = (0g𝐹)))
1331323adant3 1132 . . . . . . . . . 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 2820 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ ⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅)))))
135 opelxp 5705 . . . . . . . . . . . . 13 (⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅))))
136134, 135bitrdi 286 . . . . . . . . . . . 12 (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))))
1371363ad2ant3 1135 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g𝑅)) × (𝑌 supp (0g𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g𝑅)))))
138 simp2l 1199 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑎𝐴)
139 simp2r 1200 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑏𝐴)
140 eqidd 2732 . . . . . . . . . . . . . 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 1138 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐴𝑏𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
14217mptex 7209 . . . . . . . . . . . . . . 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 7552 . . . . . . . . . . . . 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 1463 . . . . . . . . . . . 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 2733 . . . . . . . . . . 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 917 . . . . . . . . . 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 256 . . . . . . . . 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 1411 . . . . . . . 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 1121 . . . . . . 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 2211 . . . . . 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 2211 . . . . 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 42730 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅)))) finSupp (0g𝐹))
1552, 13, 16, 19, 51, 154gsumcl 19742 . 2 (𝜑 → (𝐹 Σg (𝑎𝐴, 𝑏𝐴 ↦ (𝑖𝐴 ↦ if(𝑖 = (𝑎(+g𝑀)𝑏), ((𝑋𝑎)(.r𝑅)(𝑌𝑏)), (0g𝑅))))) ∈ 𝐵)
15612, 155eqeltrd 2832 1 (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2939  wral 3060  Vcvv 3473  ifcif 4522  {csn 4622  cop 4628   class class class wbr 5141  cmpt 5224   × cxp 5667   Fn wfn 6527  wf 6528  cfv 6532  (class class class)co 7393  cmpo 7395   supp csupp 8128  m cmap 8803  Fincfn 8922   finSupp cfsupp 9344  Basecbs 17126  +gcplusg 17179  .rcmulr 17180  0gc0g 17367   Σg cgsu 17368  CMndccmn 19612  Ringcrg 20014  LModclmod 20420   MndRing cmnring 42734
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-supp 8129  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-map 8805  df-ixp 8875  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-fsupp 9345  df-sup 9419  df-oi 9487  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-3 12258  df-4 12259  df-5 12260  df-6 12261  df-7 12262  df-8 12263  df-9 12264  df-n0 12455  df-z 12541  df-dec 12660  df-uz 12805  df-fz 13467  df-fzo 13610  df-seq 13949  df-hash 14273  df-struct 17062  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-hom 17203  df-cco 17204  df-0g 17369  df-gsum 17370  df-prds 17375  df-pws 17377  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-grp 18797  df-minusg 18798  df-sbg 18799  df-subg 18975  df-cntz 19147  df-cmn 19614  df-abl 19615  df-mgp 19947  df-ur 19964  df-ring 20016  df-subrg 20310  df-lmod 20422  df-lss 20492  df-sra 20734  df-rgmod 20735  df-dsmm 21220  df-frlm 21235  df-mnring 42735
This theorem is referenced by: (None)
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