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Theorem elrspunsn 33437
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 2735 . . . . . 6 (LIdeal‘𝑅) = (LIdeal‘𝑅)
82, 7lidlss 21240 . . . . 5 (𝐼 ∈ (LIdeal‘𝑅) → 𝐼𝐵)
96, 8syl 17 . . . 4 (𝜑𝐼𝐵)
10 elrspunsn.x . . . . . 6 (𝜑𝑋 ∈ (𝐵𝐼))
1110eldifad 3975 . . . . 5 (𝜑𝑋𝐵)
1211snssd 4814 . . . 4 (𝜑 → {𝑋} ⊆ 𝐵)
139, 12unssd 4202 . . 3 (𝜑 → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
141, 2, 3, 4, 5, 13elrsp 33380 . 2 (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥))))))
15 oveq1 7438 . . . . . . . 8 (𝑟 = (𝑎𝑋) → (𝑟 · 𝑋) = ((𝑎𝑋) · 𝑋))
1615oveq1d 7446 . . . . . . 7 (𝑟 = (𝑎𝑋) → ((𝑟 · 𝑋) + 𝑖) = (((𝑎𝑋) · 𝑋) + 𝑖))
1716eqeq2d 2746 . . . . . 6 (𝑟 = (𝑎𝑋) → (𝐴 = ((𝑟 · 𝑋) + 𝑖) ↔ 𝐴 = (((𝑎𝑋) · 𝑋) + 𝑖)))
18 oveq2 7439 . . . . . . 7 (𝑖 = (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) → (((𝑎𝑋) · 𝑋) + 𝑖) = (((𝑎𝑋) · 𝑋) + (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)))))
1918eqeq2d 2746 . . . . . 6 (𝑖 = (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) → (𝐴 = (((𝑎𝑋) · 𝑋) + 𝑖) ↔ 𝐴 = (((𝑎𝑋) · 𝑋) + (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))))))
20 elmapi 8888 . . . . . . . 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 4665 . . . . . . . 8 (𝑋𝐵𝑋 ∈ {𝑋})
24 elun2 4193 . . . . . . . 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 7105 . . . . . 6 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑎𝑋) ∈ 𝐵)
272fvexi 6921 . . . . . . . . . . 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 4188 . . . . . . . . . . . 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 6776 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑎𝐼):𝐼𝐵)
3328, 29, 32elmapdd 8880 . . . . . . . . 9 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑎𝐼) ∈ (𝐵m 𝐼))
34 breq1 5151 . . . . . . . . . . 11 (𝑏 = (𝑎𝐼) → (𝑏 finSupp 0 ↔ (𝑎𝐼) finSupp 0 ))
35 fveq1 6906 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎𝐼) → (𝑏𝑦) = ((𝑎𝐼)‘𝑦))
3635oveq1d 7446 . . . . . . . . . . . . . 14 (𝑏 = (𝑎𝐼) → ((𝑏𝑦) · 𝑦) = (((𝑎𝐼)‘𝑦) · 𝑦))
3736mpteq2dv 5250 . . . . . . . . . . . . 13 (𝑏 = (𝑎𝐼) → (𝑦𝐼 ↦ ((𝑏𝑦) · 𝑦)) = (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦)))
3837oveq2d 7447 . . . . . . . . . . . 12 (𝑏 = (𝑎𝐼) → (𝑅 Σg (𝑦𝐼 ↦ ((𝑏𝑦) · 𝑦))) = (𝑅 Σg (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦))))
3938eqeq2d 2746 . . . . . . . . . . 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 769 . . . . . . . . . . 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 20281 . . . . . . . . . . . 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 9431 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑎𝐼) finSupp 0 )
47 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (𝑎𝑥) = (𝑎𝑦))
48 id 22 . . . . . . . . . . . . . 14 (𝑥 = 𝑦𝑥 = 𝑦)
4947, 48oveq12d 7449 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑎𝑥) · 𝑥) = ((𝑎𝑦) · 𝑦))
5049cbvmptv 5261 . . . . . . . . . . . 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 6927 . . . . . . . . . . . . . 14 (((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) ∧ 𝑦𝐼) → ((𝑎𝐼)‘𝑦) = (𝑎𝑦))
5352oveq1d 7446 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) ∧ 𝑦𝐼) → (((𝑎𝐼)‘𝑦) · 𝑦) = ((𝑎𝑦) · 𝑦))
5453mpteq2dva 5248 . . . . . . . . . . . 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 2794 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))) → (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)) = (𝑦𝐼 ↦ (((𝑎𝐼)‘𝑦) · 𝑦)))
5655oveq2d 7447 . . . . . . . . . 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 3624 . . . . . . . 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 33380 . . . . . . . 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 33383 . . . . . . . . 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 2841 . . . . . 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 20298 . . . . . . . . . . 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 5442 . . . . . . . . . . . 12 {𝑋} ∈ V
7271a1i 11 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → {𝑋} ∈ V)
7370, 72unexd 7773 . . . . . . . . . 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 7105 . . . . . . . . . . 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 3996 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥𝐵)
802, 4, 74, 77, 79ringcld 20277 . . . . . . . . . 10 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → ((𝑎𝑥) · 𝑥) ∈ 𝐵)
8173mptexd 7244 . . . . . . . . . . 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 6606 . . . . . . . . . . . 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 9407 . . . . . . . . . . . 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 6738 . . . . . . . . . . . . . . . 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 8195 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → (𝑎𝑥) = 0 )
9594oveq1d 7446 . . . . . . . . . . . . . 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 3975 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
9896, 97sseldd 3996 . . . . . . . . . . . . . . 15 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑥𝐵)
992, 4, 3ringlz 20307 . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → ((𝑎𝑥) · 𝑥) = 0 )
102101, 73suppss2 8224 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)) supp 0 ) ⊆ (𝑎 supp 0 ))
10387, 102ssfid 9299 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)) supp 0 ) ∈ Fin)
10481, 83, 85, 103isfsuppd 9404 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)) finSupp 0 )
10510eldifbd 3976 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑋𝐼)
106105ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ¬ 𝑋𝐼)
107 disjsn 4716 . . . . . . . . . . 11 ((𝐼 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐼)
108106, 107sylibr 234 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∩ {𝑋}) = ∅)
109 eqidd 2736 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∪ {𝑋}) = (𝐼 ∪ {𝑋}))
1102, 3, 67, 69, 73, 80, 104, 108, 109gsumsplit2 19962 . . . . . . . . 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 19837 . . . . . . . . . . 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 4189 . . . . . . . . . . . . . 14 {𝑋} ⊆ (𝐼 ∪ {𝑋})
11611, 23syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 ∈ {𝑋})
117116ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑋 ∈ {𝑋})
118115, 117sselid 3993 . . . . . . . . . . . . 13 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑋 ∈ (𝐼 ∪ {𝑋}))
119114, 118ffvelcdmd 7105 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑎𝑋) ∈ 𝐵)
1202, 4, 113, 119, 112ringcld 20277 . . . . . . . . . . 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 6911 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 = 𝑋) → (𝑎𝑥) = (𝑎𝑋))
123122, 121oveq12d 7449 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 = 𝑋) → ((𝑎𝑥) · 𝑥) = ((𝑎𝑋) · 𝑋))
1242, 111, 112, 120, 123gsumsnd 19985 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σg (𝑥 ∈ {𝑋} ↦ ((𝑎𝑥) · 𝑥))) = ((𝑎𝑋) · 𝑋))
125124oveq2d 7447 . . . . . . . . 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 3993 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
130127, 129ffvelcdmd 7105 . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → 𝑥𝐵)
1332, 4, 126, 130, 132ringcld 20277 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥𝐼) → ((𝑎𝑥) · 𝑥) ∈ 𝐵)
134133fmpttd 7135 . . . . . . . . . . 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 4155 . . . . . . . . . . . . . . . . 17 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∖ (𝑎 supp 0 )) ⊆ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 )))
137136sselda 3995 . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . 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 3975 . . . . . . . . . . . . . . . 16 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑥𝐼)
144141, 143sseldd 3996 . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → ((𝑎𝑥) · 𝑥) = 0 )
147146, 70suppss2 8224 . . . . . . . . . . . 12 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)) supp 0 ) ⊆ (𝑎 supp 0 ))
14887, 147ssfid 9299 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥)) supp 0 ) ∈ Fin)
1492, 3, 69, 70, 134, 148gsumcl2 19947 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σg (𝑥𝐼 ↦ ((𝑎𝑥) · 𝑥))) ∈ 𝐵)
1502, 67cmncom 19831 . . . . . . . . . 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 1370 . . . . . . . . 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 2779 . . . . . . . 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 2775 . . . . . 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 3636 . . . . 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 3156 . . 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 7773 . . . . . 6 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝐼 ∪ {𝑋}) ∈ V)
162 simp-4r 784 . . . . . . . . 9 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → 𝑟𝐵)
163 eqid 2735 . . . . . . . . . . . 12 (1r𝑅) = (1r𝑅)
1642, 163ringidcl 20280 . . . . . . . . . . 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 4577 . . . . . . . 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 4577 . . . . . . 7 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1r𝑅)), 0 ) ∈ 𝐵)
170169fmpttd 7135 . . . . . 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 8880 . . . . 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 5151 . . . . . . 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 6906 . . . . . . . . . . 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 7446 . . . . . . . . . 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 5250 . . . . . . . . 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 7447 . . . . . . . 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 2746 . . . . . . 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 2735 . . . . . . 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 9361 . . . . . . . 8 {𝑋, 𝑖} ∈ Fin
182181a1i 11 . . . . . . 7 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → {𝑋, 𝑖} ∈ Fin)
183 simp-4r 784 . . . . . . . 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 4577 . . . . . . 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 32708 . . . . . 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 769 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖𝐼)
191189, 190sseldd 3996 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖𝐵)
1925ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑅 ∈ Ring)
193 simpllr 776 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑟𝐵)
19411ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑋𝐵)
1952, 4, 192, 193, 194ringcld 20277 . . . . . . . . 9 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑟 · 𝑋) ∈ 𝐵)
1962, 67cmncom 19831 . . . . . . . . 9 ((𝑅 ∈ CMnd ∧ 𝑖𝐵 ∧ (𝑟 · 𝑋) ∈ 𝐵) → (𝑖 + (𝑟 · 𝑋)) = ((𝑟 · 𝑋) + 𝑖))
197188, 191, 195, 196syl3anc 1370 . . . . . . . 8 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑖 + (𝑟 · 𝑋)) = ((𝑟 · 𝑋) + 𝑖))
198188cmnmndd 19837 . . . . . . . . . . 11 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑅 ∈ Mnd)
199 eqid 2735 . . . . . . . . . . 11 (𝑥𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 )) = (𝑥𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 ))
200191, 2eleqtrdi 2849 . . . . . . . . . . 11 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖 ∈ (Base‘𝑅))
2013, 198, 159, 190, 199, 200gsummptif1n0 19999 . . . . . . . . . 10 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑅 Σg (𝑥𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 ))) = 𝑖)
202 fveq2 6907 . . . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . . . 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 4766 . . . . . . . . . . . . . . . . . . . 20 (𝑖𝐼𝑖 ∈ {𝑋, 𝑖})
208207ad5antlr 735 . . . . . . . . . . . . . . . . . . 19 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑖 ∈ {𝑋, 𝑖})
209206, 208eqeltrd 2839 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑦 ∈ {𝑋, 𝑖})
210209iftrued 4539 . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . . . 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 2863 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝐼 ∧ ¬ 𝑋𝐼) → ¬ 𝑦 = 𝑋)
217213, 215, 216syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → ¬ 𝑦 = 𝑋)
218217iffalsed 4542 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → if(𝑦 = 𝑋, 𝑟, (1r𝑅)) = (1r𝑅))
219210, 218eqtrd 2775 . . . . . . . . . . . . . . . 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 3993 . . . . . . . . . . . . . . . 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 7024 . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . 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 20286 . . . . . . . . . . . . . 14 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ 𝑥 = 𝑖) → ((1r𝑅) · 𝑖) = 𝑖)
227205, 224, 2263eqtrrd 2780 . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → (𝑦 ∈ {𝑋, 𝑖} ↔ 𝑥 ∈ {𝑋, 𝑖}))
229 eqeq1 2739 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → (𝑦 = 𝑋𝑥 = 𝑋))
230229ifbid 4554 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → if(𝑦 = 𝑋, 𝑟, (1r𝑅)) = if(𝑥 = 𝑋, 𝑟, (1r𝑅)))
231228, 230ifbieq1d 4555 . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥𝐼)
23330, 232sselid 3993 . . . . . . . . . . . . . . . . 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 4577 . . . . . . . . . . . . . . . . . 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 4577 . . . . . . . . . . . . . . . . 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 7039 . . . . . . . . . . . . . . . 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 3038 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐼 ∧ ¬ 𝑋𝐼) → 𝑥𝑋)
242232, 240, 241syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥𝑋)
243 neqne 2946 . . . . . . . . . . . . . . . . . . 19 𝑥 = 𝑖𝑥𝑖)
244243adantl 481 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥𝑖)
245242, 244nelprd 4662 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → ¬ 𝑥 ∈ {𝑋, 𝑖})
246245iffalsed 4542 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥𝐼) ∧ ¬ 𝑥 = 𝑖) → if(𝑥 ∈ {𝑋, 𝑖}, if(𝑥 = 𝑋, 𝑟, (1r𝑅)), 0 ) = 0 )
247239, 246eqtrd 2775 . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . 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 4567 . . . . . . . . . . . 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 5248 . . . . . . . . . . 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 7447 . . . . . . . . . 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 2777 . . . . . . . . 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 769 . . . . . . . . . . . . . . . . 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 4765 . . . . . . . . . . . . . . . . . 18 (𝑋𝐵𝑋 ∈ {𝑋, 𝑖})
262260, 261syl 17 . . . . . . . . . . . . . . . . 17 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑋 ∈ {𝑋, 𝑖})
263259, 262eqeltrd 2839 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑥 ∈ {𝑋, 𝑖})
264258, 263eqeltrd 2839 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑦 ∈ {𝑋, 𝑖})
265264iftrued 4539 . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . 15 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑋)
267266iftrued 4539 . . . . . . . . . . . . . 14 ((((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → if(𝑦 = 𝑋, 𝑟, (1r𝑅)) = 𝑟)
268265, 267eqtrd 2775 . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . 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 7024 . . . . . . . . . . . 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 7449 . . . . . . . . . . 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 19985 . . . . . . . . . 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 2741 . . . . . . . . 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 7449 . . . . . . . 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 2777 . . . . . . 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 7104 . . . . . . . . 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 3996 . . . . . . . . 9 (((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥𝐵)
2862, 4, 281, 282, 285ringcld 20277 . . . . . . . 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 7244 . . . . . . . . 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 6606 . . . . . . . . . 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 9407 . . . . . . . . . 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 1912 . . . . . . . . . . . . . . . 16 𝑦(((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖))
292291, 169, 180fnmptd 6710 . . . . . . . . . . . . . . 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 8195 . . . . . . . . . . . . 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 7446 . . . . . . . . . . . 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 3975 . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 8224 . . . . . . . . . 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 9299 . . . . . . . . 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 9404 . . . . . . . 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 2736 . . . . . . . 8 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝐼 ∪ {𝑋}) = (𝐼 ∪ {𝑋}))
3102, 3, 67, 188, 161, 286, 307, 308, 309gsumsplit2 19962 . . . . . . 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 2785 . . . . . 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 3624 . . . 4 ((((𝜑𝑟𝐵) ∧ 𝑖𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ∃𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))))
314313r19.29ffa 32500 . . 3 ((𝜑 ∧ ∃𝑟𝐵𝑖𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ∃𝑎 ∈ (𝐵m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0𝐴 = (𝑅 Σg (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎𝑥) · 𝑥)))))
315157, 314impbida 801 . 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 1537  wcel 2106  wne 2938  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  ifcif 4531  {csn 4631  {cpr 4633   class class class wbr 5148  cmpt 5231  cres 5691  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431   supp csupp 8184  m cmap 8865  Fincfn 8984   finSupp cfsupp 9399  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  0gc0g 17486   Σg cgsu 17487  CMndccmn 19813  1rcur 20199  Ringcrg 20251  LIdealclidl 21234  RSpancrsp 21235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  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 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-nzr 20530  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lbs 21092  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-dsmm 21770  df-frlm 21785  df-uvc 21821
This theorem is referenced by:  qsdrngilem  33502
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