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Theorem elrspunsn 33458
Description: Membership to the span of an ideal 𝑅 and a single element 𝑋. (Contributed by Thierry Arnoux, 9-Mar-2025.)
Hypotheses
Ref Expression
elrspunidl.n 𝑁 = (RSpan‘𝑅)
elrspunidl.b 𝐵 = (Base‘𝑅)
elrspunidl.1 0 = (0g𝑅)
elrspunidl.x · = (.r𝑅)
elrspunidl.r (𝜑𝑅 ∈ Ring)
elrspunsn.p + = (+g𝑅)
elrspunsn.i (𝜑𝐼 ∈ (LIdeal‘𝑅))
elrspunsn.x (𝜑𝑋 ∈ (𝐵𝐼))
Assertion
Ref Expression
elrspunsn (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑟𝐵𝑖𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)))
Distinct variable groups:   0 ,𝑖   · ,𝑖   𝐵,𝑖   𝑅,𝑖   𝑖,𝑋   𝜑,𝑖   + ,𝑖,𝑟   0 ,𝑟   · ,𝑟   𝐴,𝑖,𝑟   𝐵,𝑟   𝑖,𝐼,𝑟   𝑅,𝑟   𝑋,𝑟   𝜑,𝑟
Allowed substitution hints:   𝑁(𝑖,𝑟)

Proof of Theorem elrspunsn
Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elrspunidl.n . . 3 𝑁 = (RSpan‘𝑅)
2 elrspunidl.b . . 3 𝐵 = (Base‘𝑅)
3 elrspunidl.1 . . 3 0 = (0g𝑅)
4 elrspunidl.x . . 3 · = (.r𝑅)
5 elrspunidl.r . . 3 (𝜑𝑅 ∈ Ring)
6 elrspunsn.i . . . . 5 (𝜑𝐼 ∈ (LIdeal‘𝑅))
7 eqid 2736 . . . . . 6 (LIdeal‘𝑅) = (LIdeal‘𝑅)
82, 7lidlss 21223 . . . . 5 (𝐼 ∈ (LIdeal‘𝑅) → 𝐼𝐵)
96, 8syl 17 . . . 4 (𝜑𝐼𝐵)
10 elrspunsn.x . . . . . 6 (𝜑𝑋 ∈ (𝐵𝐼))
1110eldifad 3962 . . . . 5 (𝜑𝑋𝐵)
1211snssd 4808 . . . 4 (𝜑 → {𝑋} ⊆ 𝐵)
139, 12unssd 4191 . . 3 (𝜑 → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
141, 2, 3, 4, 5, 13elrsp 33401 . 2 (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))))))
15 oveq1 7439 . . . . . . . 8 (𝑟 = (𝑎𝑋) → (𝑟 · 𝑋) = ((𝑎𝑋) · 𝑋))
1615oveq1d 7447 . . . . . . 7 (𝑟 = (𝑎𝑋) → ((𝑟 · 𝑋) + 𝑖) = (((𝑎𝑋) · 𝑋) + 𝑖))
1716eqeq2d 2747 . . . . . 6 (𝑟 = (𝑎𝑋) → (𝐴 = ((𝑟 · 𝑋) + 𝑖) ↔ 𝐴 = (((𝑎𝑋) · 𝑋) + 𝑖)))
18 oveq2 7440 . . . . . . 7 (𝑖 = (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) → (((𝑎𝑋) · 𝑋) + 𝑖) = (((𝑎𝑋) · 𝑋) + (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)))))
1918eqeq2d 2747 . . . . . 6 (𝑖 = (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) → (𝐴 = (((𝑎𝑋) · 𝑋) + 𝑖) ↔ 𝐴 = (((𝑎𝑋) · 𝑋) + (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))))))
20 elmapi 8890 . . . . . . . 8 (𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋})) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
2120ad3antlr 731 . . . . . . 7 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
2211ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝑋𝐵)
23 snidg 4659 . . . . . . . 8 (𝑋𝐵𝑋 ∈ {𝑋})
24 elun2 4182 . . . . . . . 8 (𝑋 ∈ {𝑋} → 𝑋 ∈ (𝐼 ∪ {𝑋}))
2522, 23, 243syl 18 . . . . . . 7 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝑋 ∈ (𝐼 ∪ {𝑋}))
2621, 25ffvelcdmd 7104 . . . . . 6 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑎𝑋) ∈ 𝐵)
272fvexi 6919 . . . . . . . . . . 11 𝐵 ∈ V
2827a1i 11 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝐵 ∈ V)
296ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝐼 ∈ (LIdeal‘𝑅))
30 ssun1 4177 . . . . . . . . . . . 12 𝐼 ⊆ (𝐼 ∪ {𝑋})
3130a1i 11 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝐼 ⊆ (𝐼 ∪ {𝑋}))
3221, 31fssresd 6774 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑎𝐼):𝐼𝐵)
3328, 29, 32elmapdd 8882 . . . . . . . . 9 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑎𝐼) ∈ (𝐵m 𝐼))
34 breq1 5145 . . . . . . . . . . 11 (𝑏 = (𝑎𝐼) → (𝑏 finSupp 0 ↔ (𝑎𝐼) finSupp 0 ))
35 fveq1 6904 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎𝐼) → (𝑏𝑦) = ((𝑎𝐼)‘𝑦))
3635oveq1d 7447 . . . . . . . . . . . . . 14 (𝑏 = (𝑎𝐼) → ((𝑏𝑦) · 𝑦) = (((𝑎𝐼)‘𝑦) · 𝑦))
3736mpteq2dv 5243 . . . . . . . . . . . . 13 (𝑏 = (𝑎𝐼) → (𝑦𝐼 ↦ ((𝑏𝑦) · 𝑦)) = (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦)))
3837oveq2d 7448 . . . . . . . . . . . 12 (𝑏 = (𝑎𝐼) → (𝑅 Σg (𝑦𝐼 ↦ ((𝑏𝑦) · 𝑦))) = (𝑅 Σg (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦))))
3938eqeq2d 2747 . . . . . . . . . . 11 (𝑏 = (𝑎𝐼) → ((𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ ((𝑏𝑦) · 𝑦))) ↔ (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦)))))
4034, 39anbi12d 632 . . . . . . . . . 10 (𝑏 = (𝑎𝐼) → ((𝑏 finSupp 0 ∧ (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ ((𝑏𝑦) · 𝑦)))) ↔ ((𝑎𝐼) finSupp 0 ∧ (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦))))))
4140adantl 481 . . . . . . . . 9 (((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) ∧ 𝑏 = (𝑎𝐼)) → ((𝑏 finSupp 0 ∧ (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ ((𝑏𝑦) · 𝑦)))) ↔ ((𝑎𝐼) finSupp 0 ∧ (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦))))))
42 simplr 768 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝑎 finSupp 0 )
435ad3antrrr 730 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝑅 ∈ Ring)
442, 3ring0cl 20265 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 0𝐵)
4543, 44syl 17 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 0𝐵)
4642, 45fsuppres 9434 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑎𝐼) finSupp 0 )
47 fveq2 6905 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑎𝑥) = (𝑎𝑦))
48 id 22 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝑥 = 𝑦)
4947, 48oveq12d 7450 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑎𝑥) · 𝑥) = ((𝑎𝑦) · 𝑦))
5049cbvmptv 5254 . . . . . . . . . . . 12 (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)) = (𝑦𝐼 ↦ ((𝑎𝑦) · 𝑦))
51 simpr 484 . . . . . . . . . . . . . . 15 (((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) ∧ 𝑦𝐼) → 𝑦𝐼)
5251fvresd 6925 . . . . . . . . . . . . . 14 (((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) ∧ 𝑦𝐼) → ((𝑎𝐼)‘𝑦) = (𝑎𝑦))
5352oveq1d 7447 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) ∧ 𝑦𝐼) → (((𝑎𝐼)‘𝑦) · 𝑦) = ((𝑎𝑦) · 𝑦))
5453mpteq2dva 5241 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦)) = (𝑦𝐼 ↦ ((𝑎𝑦) · 𝑦)))
5550, 54eqtr4id 2795 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)) = (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦)))
5655oveq2d 7448 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦))))
5746, 56jca 511 . . . . . . . . 9 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → ((𝑎𝐼) finSupp 0 ∧ (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦)))))
5833, 41, 57rspcedvd 3623 . . . . . . . 8 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → ∃𝑏 ∈ (𝐵m 𝐼)(𝑏 finSupp 0 ∧ (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ ((𝑏𝑦) · 𝑦)))))
599ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝐼𝐵)
601, 2, 3, 4, 43, 59elrsp 33401 . . . . . . . 8 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → ((𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) ∈ (𝑁𝐼) ↔ ∃𝑏 ∈ (𝐵m 𝐼)(𝑏 finSupp 0 ∧ (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑦𝐼 ↦ ((𝑏𝑦) · 𝑦))))))
6158, 60mpbird 257 . . . . . . 7 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) ∈ (𝑁𝐼))
621, 7rspidlid 33404 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑁𝐼) = 𝐼)
635, 6, 62syl2anc 584 . . . . . . . 8 (𝜑 → (𝑁𝐼) = 𝐼)
6463ad3antrrr 730 . . . . . . 7 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑁𝐼) = 𝐼)
6561, 64eleqtrd 2842 . . . . . 6 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) ∈ 𝐼)
66 simpr 484 . . . . . . 7 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))))
67 elrspunsn.p . . . . . . . . . 10 + = (+g𝑅)
685ringcmnd 20282 . . . . . . . . . . 11 (𝜑𝑅 ∈ CMnd)
6968ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑅 ∈ CMnd)
706ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝐼 ∈ (LIdeal‘𝑅))
71 snex 5435 . . . . . . . . . . . 12 {𝑋} ∈ V
7271a1i 11 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → {𝑋} ∈ V)
7370, 72unexd 7775 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∪ {𝑋}) ∈ V)
745ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑅 ∈ Ring)
7520ad3antlr 731 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
76 simpr 484 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
7775, 76ffvelcdmd 7104 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → (𝑎𝑥) ∈ 𝐵)
7813ad3antrrr 730 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
7978, 76sseldd 3983 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥𝐵)
802, 4, 74, 77, 79ringcld 20258 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → ((𝑎𝑥) · 𝑥) ∈ 𝐵)
8173mptexd 7245 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)) ∈ V)
825, 44syl 17 . . . . . . . . . . . 12 (𝜑0𝐵)
8382ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 0𝐵)
84 funmpt 6603 . . . . . . . . . . . 12 Fun (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))
8584a1i 11 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → Fun (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))
86 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑎 finSupp 0 )
8786fsuppimpd 9410 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑎 supp 0 ) ∈ Fin)
8820ad3antlr 731 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
8988ffnd 6736 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑎 Fn (𝐼 ∪ {𝑋}))
9073adantr 480 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → (𝐼 ∪ {𝑋}) ∈ V)
915ad3antrrr 730 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑅 ∈ Ring)
9291, 44syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 0𝐵)
93 simpr 484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 )))
9489, 90, 92, 93fvdifsupp 8197 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → (𝑎𝑥) = 0 )
9594oveq1d 7447 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → ((𝑎𝑥) · 𝑥) = ( 0 · 𝑥))
9613ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
9793eldifad 3962 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
9896, 97sseldd 3983 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑥𝐵)
992, 4, 3ringlz 20291 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑥𝐵) → ( 0 · 𝑥) = 0 )
10091, 98, 99syl2anc 584 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → ( 0 · 𝑥) = 0 )
10195, 100eqtrd 2776 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → ((𝑎𝑥) · 𝑥) = 0 )
102101, 73suppss2 8226 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)) supp 0 ) ⊆ (𝑎 supp 0 ))
10387, 102ssfid 9302 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)) supp 0 ) ∈ Fin)
10481, 83, 85, 103isfsuppd 9407 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)) finSupp 0 )
10510eldifbd 3963 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑋𝐼)
106105ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ¬ 𝑋𝐼)
107 disjsn 4710 . . . . . . . . . . 11 ((𝐼 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐼)
108106, 107sylibr 234 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∩ {𝑋}) = ∅)
109 eqidd 2737 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∪ {𝑋}) = (𝐼 ∪ {𝑋}))
1102, 3, 67, 69, 73, 80, 104, 108, 109gsumsplit2 19948 . . . . . . . . 9 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))) = ((𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) + (𝑅 Σg (𝑥 ∈ {𝑋} ↦ ((𝑎𝑥) · 𝑥)))))
11169cmnmndd 19823 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑅 ∈ Mnd)
11211ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑋𝐵)
1135ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑅 ∈ Ring)
11420ad2antlr 727 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
115 ssun2 4178 . . . . . . . . . . . . . 14 {𝑋} ⊆ (𝐼 ∪ {𝑋})
11611, 23syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 ∈ {𝑋})
117116ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑋 ∈ {𝑋})
118115, 117sselid 3980 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑋 ∈ (𝐼 ∪ {𝑋}))
119114, 118ffvelcdmd 7104 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑎𝑋) ∈ 𝐵)
1202, 4, 113, 119, 112ringcld 20258 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑎𝑋) · 𝑋) ∈ 𝐵)
121 simpr 484 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
122121fveq2d 6909 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 = 𝑋) → (𝑎𝑥) = (𝑎𝑋))
123122, 121oveq12d 7450 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 = 𝑋) → ((𝑎𝑥) · 𝑥) = ((𝑎𝑋) · 𝑋))
1242, 111, 112, 120, 123gsumsnd 19971 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σg (𝑥 ∈ {𝑋} ↦ ((𝑎𝑥) · 𝑥))) = ((𝑎𝑋) · 𝑋))
125124oveq2d 7448 . . . . . . . . 9 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) + (𝑅 Σg (𝑥 ∈ {𝑋} ↦ ((𝑎𝑥) · 𝑥)))) = ((𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) + ((𝑎𝑋) · 𝑋)))
1265ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → 𝑅 ∈ Ring)
12720ad3antlr 731 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
128 simpr 484 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → 𝑥𝐼)
12930, 128sselid 3980 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
130127, 129ffvelcdmd 7104 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → (𝑎𝑥) ∈ 𝐵)
1319ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → 𝐼𝐵)
132131, 128sseldd 3983 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → 𝑥𝐵)
1332, 4, 126, 130, 132ringcld 20258 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → ((𝑎𝑥) · 𝑥) ∈ 𝐵)
134133fmpttd 7134 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)):𝐼𝐵)
13530a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝐼 ⊆ (𝐼 ∪ {𝑋}))
136135ssdifd 4144 . . . . . . . . . . . . . . . . 17 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∖ (𝑎 supp 0 )) ⊆ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 )))
137136sselda 3982 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 )))
138137, 94syldan 591 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → (𝑎𝑥) = 0 )
139138oveq1d 7447 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → ((𝑎𝑥) · 𝑥) = ( 0 · 𝑥))
1405ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑅 ∈ Ring)
1419ad3antrrr 730 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝐼𝐵)
142 simpr 484 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 )))
143142eldifad 3962 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑥𝐼)
144141, 143sseldd 3983 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑥𝐵)
145140, 144, 99syl2anc 584 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → ( 0 · 𝑥) = 0 )
146139, 145eqtrd 2776 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → ((𝑎𝑥) · 𝑥) = 0 )
147146, 70suppss2 8226 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)) supp 0 ) ⊆ (𝑎 supp 0 ))
14887, 147ssfid 9302 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)) supp 0 ) ∈ Fin)
1492, 3, 69, 70, 134, 148gsumcl2 19933 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) ∈ 𝐵)
1502, 67cmncom 19817 . . . . . . . . . 10 ((𝑅 ∈ CMnd ∧ (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) ∈ 𝐵 ∧ ((𝑎𝑋) · 𝑋) ∈ 𝐵) → ((𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) + ((𝑎𝑋) · 𝑋)) = (((𝑎𝑋) · 𝑋) + (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)))))
15169, 149, 120, 150syl3anc 1372 . . . . . . . . 9 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) + ((𝑎𝑋) · 𝑋)) = (((𝑎𝑋) · 𝑋) + (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)))))
152110, 125, 1513eqtrd 2780 . . . . . . . 8 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))) = (((𝑎𝑋) · 𝑋) + (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)))))
153152adantr 480 . . . . . . 7 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))) = (((𝑎𝑋) · 𝑋) + (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)))))
15466, 153eqtrd 2776 . . . . . 6 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → 𝐴 = (((𝑎𝑋) · 𝑋) + (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)))))
15517, 19, 26, 65, 1542rspcedvdw 3635 . . . . 5 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → ∃𝑟𝐵𝑖𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖))
156155anasss 466 . . . 4 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ (𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))))) → ∃𝑟𝐵𝑖𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖))
157156r19.29an 3157 . . 3 ((𝜑 ∧ ∃𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))))) → ∃𝑟𝐵𝑖𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖))
15827a1i 11 . . . . . 6 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐵 ∈ V)
1596ad3antrrr 730 . . . . . . 7 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐼 ∈ (LIdeal‘𝑅))
16071a1i 11 . . . . . . 7 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → {𝑋} ∈ V)
161159, 160unexd 7775 . . . . . 6 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝐼 ∪ {𝑋}) ∈ V)
162 simp-4r 783 . . . . . . . . 9 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → 𝑟𝐵)
163 eqid 2736 . . . . . . . . . . . 12 (1r𝑅) = (1r𝑅)
1642, 163ringidcl 20263 . . . . . . . . . . 11 (𝑅 ∈ Ring → (1r𝑅) ∈ 𝐵)
1655, 164syl 17 . . . . . . . . . 10 (𝜑 → (1r𝑅) ∈ 𝐵)
166165ad4antr 732 . . . . . . . . 9 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → (1r𝑅) ∈ 𝐵)
167162, 166ifcld 4571 . . . . . . . 8 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → if(𝑦 = 𝑋, 𝑟, (1r𝑅)) ∈ 𝐵)
16882ad4antr 732 . . . . . . . 8 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → 0𝐵)
169167, 168ifcld 4571 . . . . . . 7 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ) ∈ 𝐵)
170169fmpttd 7134 . . . . . 6 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )):(𝐼 ∪ {𝑋})⟶𝐵)
171158, 161, 170elmapdd 8882 . . . . 5 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) ∈ (𝐵m (𝐼 ∪ {𝑋})))
172 breq1 5145 . . . . . . 7 (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) → (𝑎 finSupp 0 ↔ (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) finSupp 0 ))
173 fveq1 6904 . . . . . . . . . . 11 (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) → (𝑎𝑥) = ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥))
174173oveq1d 7447 . . . . . . . . . 10 (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) → ((𝑎𝑥) · 𝑥) = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))
175174mpteq2dv 5243 . . . . . . . . 9 (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)) = (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)))
176175oveq2d 7448 . . . . . . . 8 (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) → (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))) = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))))
177176eqeq2d 2747 . . . . . . 7 (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) → (𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))) ↔ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)))))
178172, 177anbi12d 632 . . . . . 6 (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) → ((𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) ↔ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))))))
179178adantl 481 . . . . 5 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))) → ((𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) ↔ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))))))
180 eqid 2736 . . . . . . 7 (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))
181 prfi 9364 . . . . . . . 8 {𝑋, 𝑖} ∈ Fin
182181a1i 11 . . . . . . 7 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → {𝑋, 𝑖} ∈ Fin)
183 simp-4r 783 . . . . . . . 8 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ {𝑋, 𝑖}) → 𝑟𝐵)
184165ad4antr 732 . . . . . . . 8 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ {𝑋, 𝑖}) → (1r𝑅) ∈ 𝐵)
185183, 184ifcld 4571 . . . . . . 7 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ {𝑋, 𝑖}) → if(𝑦 = 𝑋, 𝑟, (1r𝑅)) ∈ 𝐵)
18682ad3antrrr 730 . . . . . . 7 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 0𝐵)
187180, 161, 182, 185, 186mptiffisupp 32703 . . . . . 6 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) finSupp 0 )
18868ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑅 ∈ CMnd)
189159, 8syl 17 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐼𝐵)
190 simplr 768 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖𝐼)
191189, 190sseldd 3983 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖𝐵)
1925ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑅 ∈ Ring)
193 simpllr 775 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑟𝐵)
19411ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑋𝐵)
1952, 4, 192, 193, 194ringcld 20258 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑟 · 𝑋) ∈ 𝐵)
1962, 67cmncom 19817 . . . . . . . . 9 ((𝑅 ∈ CMnd ∧ 𝑖𝐵 ∧ (𝑟 · 𝑋) ∈ 𝐵) → (𝑖 + (𝑟 · 𝑋)) = ((𝑟 · 𝑋) + 𝑖))
197188, 191, 195, 196syl3anc 1372 . . . . . . . 8 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑖 + (𝑟 · 𝑋)) = ((𝑟 · 𝑋) + 𝑖))
198188cmnmndd 19823 . . . . . . . . . . 11 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑅 ∈ Mnd)
199 eqid 2736 . . . . . . . . . . 11 (𝑥𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 )) = (𝑥𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 ))
200191, 2eleqtrdi 2850 . . . . . . . . . . 11 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖 ∈ (Base‘𝑅))
2013, 198, 159, 190, 199, 200gsummptif1n0 19985 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑅 Σg (𝑥𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 ))) = 𝑖)
202 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑖 → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) = ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑖))
203 id 22 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑖𝑥 = 𝑖)
204202, 203oveq12d 7450 . . . . . . . . . . . . . . 15 (𝑥 = 𝑖 → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥) = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑖) · 𝑖))
205204adantl 481 . . . . . . . . . . . . . 14 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥) = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑖) · 𝑖))
206 simpr 484 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑦 = 𝑖)
207 prid2g 4760 . . . . . . . . . . . . . . . . . . . 20 (𝑖𝐼𝑖 ∈ {𝑋, 𝑖})
208207ad5antlr 735 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑖 ∈ {𝑋, 𝑖})
209206, 208eqeltrd 2840 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑦 ∈ {𝑋, 𝑖})
210209iftrued 4532 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ) = if(𝑦 = 𝑋, 𝑟, (1r𝑅)))
211190ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → 𝑖𝐼)
212211adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑖𝐼)
213206, 212eqeltrd 2840 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑦𝐼)
214105ad3antrrr 730 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ¬ 𝑋𝐼)
215214ad3antrrr 730 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → ¬ 𝑋𝐼)
216 nelneq 2864 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝐼 ∧ ¬ 𝑋𝐼) → ¬ 𝑦 = 𝑋)
217213, 215, 216syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → ¬ 𝑦 = 𝑋)
218217iffalsed 4535 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → if(𝑦 = 𝑋, 𝑟, (1r𝑅)) = (1r𝑅))
219210, 218eqtrd 2776 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ) = (1r𝑅))
22030, 211sselid 3980 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → 𝑖 ∈ (𝐼 ∪ {𝑋}))
221192ad2antrr 726 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → 𝑅 ∈ Ring)
222221, 164syl 17 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → (1r𝑅) ∈ 𝐵)
223180, 219, 220, 222fvmptd2 7023 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑖) = (1r𝑅))
224223oveq1d 7447 . . . . . . . . . . . . . 14 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑖) · 𝑖) = ((1r𝑅) · 𝑖))
225191ad2antrr 726 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → 𝑖𝐵)
2262, 4, 163, 221, 225ringlidmd 20270 . . . . . . . . . . . . . 14 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → ((1r𝑅) · 𝑖) = 𝑖)
227205, 224, 2263eqtrrd 2781 . . . . . . . . . . . . 13 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → 𝑖 = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))
228 eleq1w 2823 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → (𝑦 ∈ {𝑋, 𝑖} ↔ 𝑥 ∈ {𝑋, 𝑖}))
229 eqeq1 2740 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → (𝑦 = 𝑋𝑥 = 𝑋))
230229ifbid 4548 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → if(𝑦 = 𝑋, 𝑟, (1r𝑅)) = if(𝑥 = 𝑋, 𝑟, (1r𝑅)))
231228, 230ifbieq1d 4549 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ) = if(𝑥 ∈ {𝑋, 𝑖}, if(𝑥 = 𝑋, 𝑟, (1r𝑅)), 0 ))
232 simplr 768 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥𝐼)
23330, 232sselid 3980 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
234193ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑟𝐵)
235165ad5antr 734 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → (1r𝑅) ∈ 𝐵)
236234, 235ifcld 4571 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → if(𝑥 = 𝑋, 𝑟, (1r𝑅)) ∈ 𝐵)
237186ad2antrr 726 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 0𝐵)
238236, 237ifcld 4571 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → if(𝑥 ∈ {𝑋, 𝑖}, if(𝑥 = 𝑋, 𝑟, (1r𝑅)), 0 ) ∈ 𝐵)
239180, 231, 233, 238fvmptd3 7038 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) = if(𝑥 ∈ {𝑋, 𝑖}, if(𝑥 = 𝑋, 𝑟, (1r𝑅)), 0 ))
240214ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → ¬ 𝑋𝐼)
241 nelne2 3039 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐼 ∧ ¬ 𝑋𝐼) → 𝑥𝑋)
242232, 240, 241syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥𝑋)
243 neqne 2947 . . . . . . . . . . . . . . . . . . 19 𝑥 = 𝑖𝑥𝑖)
244243adantl 481 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥𝑖)
245242, 244nelprd 4656 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → ¬ 𝑥 ∈ {𝑋, 𝑖})
246245iffalsed 4535 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → if(𝑥 ∈ {𝑋, 𝑖}, if(𝑥 = 𝑋, 𝑟, (1r𝑅)), 0 ) = 0 )
247239, 246eqtrd 2776 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) = 0 )
248247oveq1d 7447 . . . . . . . . . . . . . 14 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥) = ( 0 · 𝑥))
249192ad2antrr 726 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑅 ∈ Ring)
250189ad2antrr 726 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝐼𝐵)
251250, 232sseldd 3983 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥𝐵)
252249, 251, 99syl2anc 584 . . . . . . . . . . . . . 14 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → ( 0 · 𝑥) = 0 )
253248, 252eqtr2d 2777 . . . . . . . . . . . . 13 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 0 = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))
254227, 253ifeqda 4561 . . . . . . . . . . . 12 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) → if(𝑥 = 𝑖, 𝑖, 0 ) = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))
255254mpteq2dva 5241 . . . . . . . . . . 11 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑥𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 )) = (𝑥𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)))
256255oveq2d 7448 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑅 Σg (𝑥𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 ))) = (𝑅 Σg (𝑥𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))))
257201, 256eqtr3d 2778 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖 = (𝑅 Σg (𝑥𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))))
258 simpr 484 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
259 simplr 768 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑥 = 𝑋)
260194ad2antrr 726 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑋𝐵)
261 prid1g 4759 . . . . . . . . . . . . . . . . . 18 (𝑋𝐵𝑋 ∈ {𝑋, 𝑖})
262260, 261syl 17 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑋 ∈ {𝑋, 𝑖})
263259, 262eqeltrd 2840 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑥 ∈ {𝑋, 𝑖})
264258, 263eqeltrd 2840 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑦 ∈ {𝑋, 𝑖})
265264iftrued 4532 . . . . . . . . . . . . . 14 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ) = if(𝑦 = 𝑋, 𝑟, (1r𝑅)))
266258, 259eqtrd 2776 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑋)
267266iftrued 4532 . . . . . . . . . . . . . 14 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → if(𝑦 = 𝑋, 𝑟, (1r𝑅)) = 𝑟)
268265, 267eqtrd 2776 . . . . . . . . . . . . 13 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ) = 𝑟)
269 simpr 484 . . . . . . . . . . . . . 14 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
270116ad4antr 732 . . . . . . . . . . . . . . 15 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑋 ∈ {𝑋})
271270, 24syl 17 . . . . . . . . . . . . . 14 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑋 ∈ (𝐼 ∪ {𝑋}))
272269, 271eqeltrd 2840 . . . . . . . . . . . . 13 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
273193adantr 480 . . . . . . . . . . . . 13 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑟𝐵)
274180, 268, 272, 273fvmptd2 7023 . . . . . . . . . . . 12 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) = 𝑟)
275274, 269oveq12d 7450 . . . . . . . . . . 11 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥) = (𝑟 · 𝑋))
2762, 198, 194, 195, 275gsumsnd 19971 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑅 Σg (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))) = (𝑟 · 𝑋))
277276eqcomd 2742 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑟 · 𝑋) = (𝑅 Σg (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))))
278257, 277oveq12d 7450 . . . . . . . 8 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑖 + (𝑟 · 𝑋)) = ((𝑅 Σg (𝑥𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))) + (𝑅 Σg (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)))))
279197, 278eqtr3d 2778 . . . . . . 7 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑟 · 𝑋) + 𝑖) = ((𝑅 Σg (𝑥𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))) + (𝑅 Σg (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)))))
280 simpr 484 . . . . . . 7 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐴 = ((𝑟 · 𝑋) + 𝑖))
2815ad4antr 732 . . . . . . . . 9 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑅 ∈ Ring)
282170ffvelcdmda 7103 . . . . . . . . 9 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) ∈ 𝐵)
28313ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
284 simpr 484 . . . . . . . . . 10 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
285283, 284sseldd 3983 . . . . . . . . 9 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥𝐵)
2862, 4, 281, 282, 285ringcld 20258 . . . . . . . 8 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥) ∈ 𝐵)
287161mptexd 7245 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)) ∈ V)
288 funmpt 6603 . . . . . . . . . 10 Fun (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))
289288a1i 11 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → Fun (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)))
290187fsuppimpd 9410 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ) ∈ Fin)
291 nfv 1913 . . . . . . . . . . . . . . . 16 𝑦(((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖))
292291, 169, 180fnmptd 6708 . . . . . . . . . . . . . . 15 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) Fn (𝐼 ∪ {𝑋}))
293292adantr 480 . . . . . . . . . . . . . 14 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) Fn (𝐼 ∪ {𝑋}))
294161adantr 480 . . . . . . . . . . . . . 14 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → (𝐼 ∪ {𝑋}) ∈ V)
295186adantr 480 . . . . . . . . . . . . . 14 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → 0𝐵)
296 simpr 484 . . . . . . . . . . . . . 14 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 )))
297293, 294, 295, 296fvdifsupp 8197 . . . . . . . . . . . . 13 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) = 0 )
298297oveq1d 7447 . . . . . . . . . . . 12 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥) = ( 0 · 𝑥))
2995ad4antr 732 . . . . . . . . . . . . 13 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → 𝑅 ∈ Ring)
30013ad4antr 732 . . . . . . . . . . . . . 14 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
301296eldifad 3962 . . . . . . . . . . . . . 14 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
302300, 301sseldd 3983 . . . . . . . . . . . . 13 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → 𝑥𝐵)
303299, 302, 99syl2anc 584 . . . . . . . . . . . 12 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → ( 0 · 𝑥) = 0 )
304298, 303eqtrd 2776 . . . . . . . . . . 11 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥) = 0 )
305304, 161suppss2 8226 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)) supp 0 ) ⊆ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) supp 0 ))
306290, 305ssfid 9302 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)) supp 0 ) ∈ Fin)
307287, 186, 289, 306isfsuppd 9407 . . . . . . . 8 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)) finSupp 0 )
308214, 107sylibr 234 . . . . . . . 8 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝐼 ∩ {𝑋}) = ∅)
309 eqidd 2737 . . . . . . . 8 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝐼 ∪ {𝑋}) = (𝐼 ∪ {𝑋}))
3102, 3, 67, 188, 161, 286, 307, 308, 309gsumsplit2 19948 . . . . . . 7 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))) = ((𝑅 Σg (𝑥𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))) + (𝑅 Σg (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)))))
311279, 280, 3103eqtr4d 2786 . . . . . 6 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥))))
312187, 311jca 511 . . . . 5 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 )) finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ))‘𝑥) · 𝑥)))))
313171, 179, 312rspcedvd 3623 . . . 4 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ∃𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))))
314313r19.29ffa 32491 . . 3 ((𝜑 ∧ ∃𝑟𝐵𝑖𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ∃𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))))
315157, 314impbida 800 . 2 (𝜑 → (∃𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) ↔ ∃𝑟𝐵𝑖𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)))
31614, 315bitrd 279 1 (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑟𝐵𝑖𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  wrex 3069  Vcvv 3479  cdif 3947  cun 3948  cin 3949  wss 3950  c0 4332  ifcif 4524  {csn 4625  {cpr 4627   class class class wbr 5142  cmpt 5224  cres 5686  Fun wfun 6554   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432   supp csupp 8186  m cmap 8867  Fincfn 8986   finSupp cfsupp 9402  Basecbs 17248  +gcplusg 17298  .rcmulr 17299  0gc0g 17485   Σg cgsu 17486  CMndccmn 19799  1rcur 20179  Ringcrg 20231  LIdealclidl 21217  RSpancrsp 21218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-nzr 20514  df-subrg 20571  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lmhm 21022  df-lbs 21075  df-sra 21173  df-rgmod 21174  df-lidl 21219  df-rsp 21220  df-dsmm 21753  df-frlm 21768  df-uvc 21804
This theorem is referenced by:  qsdrngilem  33523
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