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Theorem islindf4 20898
 Description: A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Hypotheses
Ref Expression
islindf4.b 𝐵 = (Base‘𝑊)
islindf4.r 𝑅 = (Scalar‘𝑊)
islindf4.t · = ( ·𝑠𝑊)
islindf4.z 0 = (0g𝑊)
islindf4.y 𝑌 = (0g𝑅)
islindf4.l 𝐿 = (Base‘(𝑅 freeLMod 𝐼))
Assertion
Ref Expression
islindf4 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0𝑥 = (𝐼 × {𝑌}))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐼   𝑥,𝐿   𝑥,𝑅   𝑥, ·   𝑥,𝑊   𝑥,𝑋   𝑥,𝑌   𝑥, 0

Proof of Theorem islindf4
Dummy variables 𝑗 𝑘 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raldifsni 4726 . . . . 5 (∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ (Base‘𝑅)((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌))
2 simpll1 1206 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ LMod)
3 simprll 775 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑙 ∈ (Base‘𝑅))
4 ffvelrn 6844 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝐼𝐵𝑗𝐼) → (𝐹𝑗) ∈ 𝐵)
543ad2antl3 1181 . . . . . . . . . . . . . . . . 17 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (𝐹𝑗) ∈ 𝐵)
65adantr 481 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝐹𝑗) ∈ 𝐵)
7 islindf4.b . . . . . . . . . . . . . . . . 17 𝐵 = (Base‘𝑊)
8 islindf4.r . . . . . . . . . . . . . . . . 17 𝑅 = (Scalar‘𝑊)
9 islindf4.t . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑊)
10 eqid 2825 . . . . . . . . . . . . . . . . 17 (invg𝑊) = (invg𝑊)
11 eqid 2825 . . . . . . . . . . . . . . . . 17 (invg𝑅) = (invg𝑅)
12 eqid 2825 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
137, 8, 9, 10, 11, 12lmodvsinv 19730 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘𝑅) ∧ (𝐹𝑗) ∈ 𝐵) → (((invg𝑅)‘𝑙) · (𝐹𝑗)) = ((invg𝑊)‘(𝑙 · (𝐹𝑗))))
142, 3, 6, 13syl3anc 1365 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((invg𝑅)‘𝑙) · (𝐹𝑗)) = ((invg𝑊)‘(𝑙 · (𝐹𝑗))))
1514eqeq1d 2827 . . . . . . . . . . . . . 14 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ ((invg𝑊)‘(𝑙 · (𝐹𝑗))) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))))
16 lmodgrp 19563 . . . . . . . . . . . . . . . 16 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
172, 16syl 17 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ Grp)
187, 8, 9, 12lmodvscl 19573 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘𝑅) ∧ (𝐹𝑗) ∈ 𝐵) → (𝑙 · (𝐹𝑗)) ∈ 𝐵)
192, 3, 6, 18syl3anc 1365 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑙 · (𝐹𝑗)) ∈ 𝐵)
20 islindf4.z . . . . . . . . . . . . . . . 16 0 = (0g𝑊)
21 lmodcmn 19604 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ LMod → 𝑊 ∈ CMnd)
222, 21syl 17 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ CMnd)
23 simpll2 1207 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐼𝑋)
24 difexg 5227 . . . . . . . . . . . . . . . . 17 (𝐼𝑋 → (𝐼 ∖ {𝑗}) ∈ V)
2523, 24syl 17 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝐼 ∖ {𝑗}) ∈ V)
26 simprlr 776 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))
27 elmapi 8421 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})) → 𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅))
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅))
29 simpll3 1208 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹:𝐼𝐵)
30 difss 4111 . . . . . . . . . . . . . . . . . 18 (𝐼 ∖ {𝑗}) ⊆ 𝐼
31 fssres 6540 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝐼𝐵 ∧ (𝐼 ∖ {𝑗}) ⊆ 𝐼) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
3229, 30, 31sylancl 586 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
338, 12, 9, 7, 2, 28, 32, 25lcomf 19595 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))):(𝐼 ∖ {𝑗})⟶𝐵)
34 islindf4.y . . . . . . . . . . . . . . . . 17 𝑌 = (0g𝑅)
35 simprr 769 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 finSupp 𝑌)
368, 12, 9, 7, 2, 28, 32, 25, 20, 34, 35lcomfsupp 19596 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))) finSupp 0 )
377, 20, 22, 25, 33, 36gsumcl 18957 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ∈ 𝐵)
38 eqid 2825 . . . . . . . . . . . . . . . 16 (+g𝑊) = (+g𝑊)
397, 38, 20, 10grpinvid2 18087 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Grp ∧ (𝑙 · (𝐹𝑗)) ∈ 𝐵 ∧ (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ∈ 𝐵) → (((invg𝑊)‘(𝑙 · (𝐹𝑗))) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ ((𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))) = 0 ))
4017, 19, 37, 39syl3anc 1365 . . . . . . . . . . . . . 14 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((invg𝑊)‘(𝑙 · (𝐹𝑗))) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ ((𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))) = 0 ))
41 simplr 765 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑗𝐼)
42 fsnunf2 6947 . . . . . . . . . . . . . . . . . . 19 ((𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) ∧ 𝑗𝐼𝑙 ∈ (Base‘𝑅)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅))
4328, 41, 3, 42syl3anc 1365 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅))
448, 12, 9, 7, 2, 43, 29, 23lcomf 19595 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹):𝐼𝐵)
45 simpr 485 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝑗𝐼)
46 simpl 483 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) → 𝑙 ∈ (Base‘𝑅))
4745, 46anim12i 612 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → (𝑗𝐼𝑙 ∈ (Base‘𝑅)))
48 elmapfun 8423 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})) → Fun 𝑦)
49 fdm 6518 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → dom 𝑦 = (𝐼 ∖ {𝑗}))
50 neldifsnd 4724 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (dom 𝑦 = (𝐼 ∖ {𝑗}) → ¬ 𝑗 ∈ (𝐼 ∖ {𝑗}))
51 df-nel 3128 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∉ dom 𝑦 ↔ ¬ 𝑗 ∈ dom 𝑦)
52 eleq2 2905 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝑦 = (𝐼 ∖ {𝑗}) → (𝑗 ∈ dom 𝑦𝑗 ∈ (𝐼 ∖ {𝑗})))
5352notbid 319 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (dom 𝑦 = (𝐼 ∖ {𝑗}) → (¬ 𝑗 ∈ dom 𝑦 ↔ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})))
5451, 53syl5bb 284 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (dom 𝑦 = (𝐼 ∖ {𝑗}) → (𝑗 ∉ dom 𝑦 ↔ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})))
5550, 54mpbird 258 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑦 = (𝐼 ∖ {𝑗}) → 𝑗 ∉ dom 𝑦)
5627, 49, 553syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})) → 𝑗 ∉ dom 𝑦)
5748, 56jca 512 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})) → (Fun 𝑦𝑗 ∉ dom 𝑦))
5857adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) → (Fun 𝑦𝑗 ∉ dom 𝑦))
5958adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → (Fun 𝑦𝑗 ∉ dom 𝑦))
6047, 59jca 512 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → ((𝑗𝐼𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦𝑗 ∉ dom 𝑦)))
61 funsnfsupp 8849 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗𝐼𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦𝑗 ∉ dom 𝑦)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌𝑦 finSupp 𝑌))
6261bicomd 224 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗𝐼𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦𝑗 ∉ dom 𝑦)) → (𝑦 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
6360, 62syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → (𝑦 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
6463biimpd 230 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → (𝑦 finSupp 𝑌 → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
6564impr 455 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌)
668, 12, 9, 7, 2, 43, 29, 23, 20, 34, 65lcomfsupp 19596 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) finSupp 0 )
67 incom 4181 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ({𝑗} ∩ (𝐼 ∖ {𝑗}))
68 disjdif 4423 . . . . . . . . . . . . . . . . . . 19 ({𝑗} ∩ (𝐼 ∖ {𝑗})) = ∅
6967, 68eqtri 2848 . . . . . . . . . . . . . . . . . 18 ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ∅
7069a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ∅)
71 difsnid 4741 . . . . . . . . . . . . . . . . . . 19 (𝑗𝐼 → ((𝐼 ∖ {𝑗}) ∪ {𝑗}) = 𝐼)
7271eqcomd 2831 . . . . . . . . . . . . . . . . . 18 (𝑗𝐼𝐼 = ((𝐼 ∖ {𝑗}) ∪ {𝑗}))
7341, 72syl 17 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐼 = ((𝐼 ∖ {𝑗}) ∪ {𝑗}))
747, 20, 38, 22, 23, 44, 66, 70, 73gsumsplit 18970 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = ((𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ (𝐼 ∖ {𝑗})))(+g𝑊)(𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ {𝑗}))))
75 vex 3502 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
76 snex 5327 . . . . . . . . . . . . . . . . . . . . 21 {⟨𝑗, 𝑙⟩} ∈ V
7775, 76unex 7461 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∈ V
78 simpl3 1187 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐹:𝐼𝐵)
79 simpl2 1186 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐼𝑋)
80 fex 6987 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:𝐼𝐵𝐼𝑋) → 𝐹 ∈ V)
8178, 79, 80syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐹 ∈ V)
8281adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹 ∈ V)
83 offres 7678 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∈ V ∧ 𝐹 ∈ V) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
8477, 82, 83sylancr 587 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
8528ffnd 6511 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 Fn (𝐼 ∖ {𝑗}))
86 neldifsn 4723 . . . . . . . . . . . . . . . . . . . . 21 ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})
87 fsnunres 6949 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 Fn (𝐼 ∖ {𝑗}) ∧ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) = 𝑦)
8885, 86, 87sylancl 586 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) = 𝑦)
8988oveq1d 7166 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘f · (𝐹 ↾ (𝐼 ∖ {𝑗}))) = (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
9084, 89eqtrd 2860 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
9190oveq2d 7167 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ (𝐼 ∖ {𝑗}))) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))))
9244ffnd 6511 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) Fn 𝐼)
93 fnressn 6915 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) Fn 𝐼𝑗𝐼) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ {𝑗}) = {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)‘𝑗)⟩})
9492, 41, 93syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ {𝑗}) = {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)‘𝑗)⟩})
9543ffnd 6511 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼)
9629ffnd 6511 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹 Fn 𝐼)
97 fnfvof 7416 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼𝐹 Fn 𝐼) ∧ (𝐼𝑋𝑗𝐼)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)‘𝑗) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹𝑗)))
9895, 96, 23, 41, 97syl22anc 836 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)‘𝑗) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹𝑗)))
99 fndm 6451 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 Fn (𝐼 ∖ {𝑗}) → dom 𝑦 = (𝐼 ∖ {𝑗}))
10099eleq2d 2902 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 Fn (𝐼 ∖ {𝑗}) → (𝑗 ∈ dom 𝑦𝑗 ∈ (𝐼 ∖ {𝑗})))
10186, 100mtbiri 328 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 Fn (𝐼 ∖ {𝑗}) → ¬ 𝑗 ∈ dom 𝑦)
102 vex 3502 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑗 ∈ V
103 vex 3502 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑙 ∈ V
104 fsnunfv 6948 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑗 ∈ V ∧ 𝑙 ∈ V ∧ ¬ 𝑗 ∈ dom 𝑦) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
105102, 103, 104mp3an12 1444 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑗 ∈ dom 𝑦 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
10685, 101, 1053syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
107106oveq1d 7166 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹𝑗)) = (𝑙 · (𝐹𝑗)))
10898, 107eqtrd 2860 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)‘𝑗) = (𝑙 · (𝐹𝑗)))
109108opeq2d 4808 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)‘𝑗)⟩ = ⟨𝑗, (𝑙 · (𝐹𝑗))⟩)
110109sneqd 4575 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)‘𝑗)⟩} = {⟨𝑗, (𝑙 · (𝐹𝑗))⟩})
111 ovex 7184 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 · (𝐹𝑗)) ∈ V
112 fmptsn 6924 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ V ∧ (𝑙 · (𝐹𝑗)) ∈ V) → {⟨𝑗, (𝑙 · (𝐹𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗))))
113102, 111, 112mp2an 688 . . . . . . . . . . . . . . . . . . . . 21 {⟨𝑗, (𝑙 · (𝐹𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))
114113a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → {⟨𝑗, (𝑙 · (𝐹𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗))))
11594, 110, 1143eqtrd 2864 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ {𝑗}) = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗))))
116115oveq2d 7167 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ {𝑗})) = (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))))
117 cmnmnd 18844 . . . . . . . . . . . . . . . . . . . 20 (𝑊 ∈ CMnd → 𝑊 ∈ Mnd)
1182, 21, 1173syl 18 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ Mnd)
119102a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑗 ∈ V)
120 eqidd 2826 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑗 → (𝑙 · (𝐹𝑗)) = (𝑙 · (𝐹𝑗)))
1217, 120gsumsn 18996 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Mnd ∧ 𝑗 ∈ V ∧ (𝑙 · (𝐹𝑗)) ∈ 𝐵) → (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))) = (𝑙 · (𝐹𝑗)))
122118, 119, 19, 121syl3anc 1365 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))) = (𝑙 · (𝐹𝑗)))
123116, 122eqtrd 2860 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ {𝑗})) = (𝑙 · (𝐹𝑗)))
12491, 123oveq12d 7169 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ (𝐼 ∖ {𝑗})))(+g𝑊)(𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹) ↾ {𝑗}))) = ((𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))))
12574, 124eqtr2d 2861 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))) = (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)))
126125eqeq1d 2827 . . . . . . . . . . . . . 14 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))) = 0 ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 ))
12715, 40, 1263bitrd 306 . . . . . . . . . . . . 13 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 ))
128106eqcomd 2831 . . . . . . . . . . . . . 14 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑙 = ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗))
129128eqeq1d 2827 . . . . . . . . . . . . 13 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑙 = 𝑌 ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))
130127, 129imbi12d 346 . . . . . . . . . . . 12 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌) ↔ ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)))
131130anassrs 468 . . . . . . . . . . 11 (((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) ∧ 𝑦 finSupp 𝑌) → (((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌) ↔ ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)))
132131pm5.74da 800 . . . . . . . . . 10 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → ((𝑦 finSupp 𝑌 → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌)) ↔ (𝑦 finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
133 impexp 451 . . . . . . . . . . 11 (((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ (𝑦 finSupp 𝑌 → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌)))
134133a1i 11 . . . . . . . . . 10 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → (((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ (𝑦 finSupp 𝑌 → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌))))
13563bicomd 224 . . . . . . . . . . 11 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌𝑦 finSupp 𝑌))
136135imbi1d 343 . . . . . . . . . 10 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)) ↔ (𝑦 finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
137132, 134, 1363bitr4d 312 . . . . . . . . 9 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗})))) → (((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
1381372ralbidva 3202 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
139 breq1 5065 . . . . . . . . . . 11 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
140 oveq1 7158 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥f · 𝐹) = ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹))
141140oveq2d 7167 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑊 Σg (𝑥f · 𝐹)) = (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)))
142141eqeq1d 2827 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑊 Σg (𝑥f · 𝐹)) = 0 ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 ))
143 fveq1 6665 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥𝑗) = ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗))
144143eqeq1d 2827 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑥𝑗) = 𝑌 ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))
145142, 144imbi12d 346 . . . . . . . . . . 11 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌) ↔ ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)))
146139, 145imbi12d 346 . . . . . . . . . 10 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)) ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
147146ralxpmap 8452 . . . . . . . . 9 (𝑗𝐼 → (∀𝑥 ∈ ((Base‘𝑅) ↑m 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
148147adantl 482 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑥 ∈ ((Base‘𝑅) ↑m 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘f · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
149138, 148bitr4d 283 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ∀𝑥 ∈ ((Base‘𝑅) ↑m 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌))))
150 breq1 5065 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 finSupp 𝑌𝑥 finSupp 𝑌))
151150ralrab 3688 . . . . . . 7 (∀𝑥 ∈ {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌} ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌) ↔ ∀𝑥 ∈ ((Base‘𝑅) ↑m 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
152149, 151syl6bbr 290 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ∀𝑥 ∈ {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌} ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
153 resima 5885 . . . . . . . . . . . . 13 ((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗})) = (𝐹 “ (𝐼 ∖ {𝑗}))
154153eqcomi 2834 . . . . . . . . . . . 12 (𝐹 “ (𝐼 ∖ {𝑗})) = ((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))
155154fveq2i 6669 . . . . . . . . . . 11 ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) = ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗})))
156155eleq2i 2908 . . . . . . . . . 10 ((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))))
157 eqid 2825 . . . . . . . . . . 11 (LSpan‘𝑊) = (LSpan‘𝑊)
15878, 30, 31sylancl 586 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
159 simpl1 1185 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝑊 ∈ LMod)
160243ad2ant2 1128 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐼 ∖ {𝑗}) ∈ V)
161160adantr 481 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (𝐼 ∖ {𝑗}) ∈ V)
162157, 7, 12, 8, 34, 9, 158, 159, 161ellspd 20862 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))) ↔ ∃𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))))))
163156, 162syl5bb 284 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∃𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗})))))))
164163imbi1d 343 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ (∃𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌)))
165 r19.23v 3283 . . . . . . . 8 (∀𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ (∃𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌))
166164, 165syl6bbr 290 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ ∀𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌)))
167166ralbidv 3201 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑m (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦f · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌)))
1688fvexi 6680 . . . . . . . . . . 11 𝑅 ∈ V
169 eqid 2825 . . . . . . . . . . . 12 (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
170 eqid 2825 . . . . . . . . . . . 12 {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌} = {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌}
171169, 12, 34, 170frlmbas 20815 . . . . . . . . . . 11 ((𝑅 ∈ V ∧ 𝐼𝑋) → {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
172168, 171mpan 686 . . . . . . . . . 10 (𝐼𝑋 → {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
1731723ad2ant2 1128 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
174173adantr 481 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
175 islindf4.l . . . . . . . 8 𝐿 = (Base‘(𝑅 freeLMod 𝐼))
176174, 175syl6reqr 2879 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐿 = {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌})
177176raleqdv 3420 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌) ↔ ∀𝑥 ∈ {𝑧 ∈ ((Base‘𝑅) ↑m 𝐼) ∣ 𝑧 finSupp 𝑌} ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
178152, 167, 1773bitr4d 312 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
1791, 178syl5bb 284 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
1808lmodfgrp 19565 . . . . . . . 8 (𝑊 ∈ LMod → 𝑅 ∈ Grp)
18112, 34, 11grpinvnzcl 18103 . . . . . . . 8 ((𝑅 ∈ Grp ∧ 𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑙) ∈ ((Base‘𝑅) ∖ {𝑌}))
182180, 181sylan 580 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑙) ∈ ((Base‘𝑅) ∖ {𝑌}))
18312, 34, 11grpinvnzcl 18103 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}))
184180, 183sylan 580 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}))
185 eldifi 4106 . . . . . . . . . 10 (𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) → 𝑘 ∈ (Base‘𝑅))
18612, 11grpinvinv 18098 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝑘 ∈ (Base‘𝑅)) → ((invg𝑅)‘((invg𝑅)‘𝑘)) = 𝑘)
187180, 185, 186syl2an 595 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘((invg𝑅)‘𝑘)) = 𝑘)
188187eqcomd 2831 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → 𝑘 = ((invg𝑅)‘((invg𝑅)‘𝑘)))
189 fveq2 6666 . . . . . . . . 9 (𝑙 = ((invg𝑅)‘𝑘) → ((invg𝑅)‘𝑙) = ((invg𝑅)‘((invg𝑅)‘𝑘)))
190189rspceeqv 3641 . . . . . . . 8 ((((invg𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}) ∧ 𝑘 = ((invg𝑅)‘((invg𝑅)‘𝑘))) → ∃𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})𝑘 = ((invg𝑅)‘𝑙))
191184, 188, 190syl2anc 584 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ∃𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})𝑘 = ((invg𝑅)‘𝑙))
192 oveq1 7158 . . . . . . . . . 10 (𝑘 = ((invg𝑅)‘𝑙) → (𝑘 · (𝐹𝑗)) = (((invg𝑅)‘𝑙) · (𝐹𝑗)))
193192eleq1d 2901 . . . . . . . . 9 (𝑘 = ((invg𝑅)‘𝑙) → ((𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
194193notbid 319 . . . . . . . 8 (𝑘 = ((invg𝑅)‘𝑙) → (¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
195194adantl 482 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑘 = ((invg𝑅)‘𝑙)) → (¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
196182, 191, 195ralxfrd 5304 . . . . . 6 (𝑊 ∈ LMod → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
1971963ad2ant1 1127 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
198197adantr 481 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
199 simplr 765 . . . . . . . 8 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → 𝑗𝐼)
20034fvexi 6680 . . . . . . . . 9 𝑌 ∈ V
201200fvconst2 6965 . . . . . . . 8 (𝑗𝐼 → ((𝐼 × {𝑌})‘𝑗) = 𝑌)
202199, 201syl 17 . . . . . . 7 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → ((𝐼 × {𝑌})‘𝑗) = 𝑌)
203202eqeq2d 2836 . . . . . 6 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → ((𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗) ↔ (𝑥𝑗) = 𝑌))
204203imbi2d 342 . . . . 5 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → (((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
205204ralbidva 3200 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
206179, 198, 2053bitr4d 312 . . 3 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
207206ralbidva 3200 . 2 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑗𝐼𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
2087, 9, 157, 8, 12, 34islindf2 20874 . 2 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑗𝐼𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
209169, 12, 175frlmbasf 20820 . . . . . . . 8 ((𝐼𝑋𝑥𝐿) → 𝑥:𝐼⟶(Base‘𝑅))
2102093ad2antl2 1180 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → 𝑥:𝐼⟶(Base‘𝑅))
211210ffnd 6511 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → 𝑥 Fn 𝐼)
212 fnconstg 6563 . . . . . . 7 (𝑌 ∈ V → (𝐼 × {𝑌}) Fn 𝐼)
213200, 212ax-mp 5 . . . . . 6 (𝐼 × {𝑌}) Fn 𝐼
214 eqfnfv 6797 . . . . . 6 ((𝑥 Fn 𝐼 ∧ (𝐼 × {𝑌}) Fn 𝐼) → (𝑥 = (𝐼 × {𝑌}) ↔ ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
215211, 213, 214sylancl 586 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → (𝑥 = (𝐼 × {𝑌}) ↔ ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
216215imbi2d 342 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → (((𝑊 Σg (𝑥f · 𝐹)) = 0𝑥 = (𝐼 × {𝑌})) ↔ ((𝑊 Σg (𝑥f · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
217216ralbidva 3200 . . 3 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0𝑥 = (𝐼 × {𝑌})) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
218 r19.21v 3179 . . . . 5 (∀𝑗𝐼 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ((𝑊 Σg (𝑥f · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
219218ralbii 3169 . . . 4 (∀𝑥𝐿𝑗𝐼 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
220 ralcom 3358 . . . 4 (∀𝑥𝐿𝑗𝐼 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
221219, 220bitr3i 278 . . 3 (∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
222217, 221syl6bb 288 . 2 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0𝑥 = (𝐼 × {𝑌})) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
223207, 208, 2223bitr4d 312 1 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥f · 𝐹)) = 0𝑥 = (𝐼 × {𝑌}))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ∉ wnel 3127  ∀wral 3142  ∃wrex 3143  {crab 3146  Vcvv 3499   ∖ cdif 3936   ∪ cun 3937   ∩ cin 3938   ⊆ wss 3939  ∅c0 4294  {csn 4563  ⟨cop 4569   class class class wbr 5062   ↦ cmpt 5142   × cxp 5551  dom cdm 5553   ↾ cres 5555   “ cima 5556  Fun wfun 6345   Fn wfn 6346  ⟶wf 6347  ‘cfv 6351  (class class class)co 7151   ∘f cof 7400   ↑m cmap 8399   finSupp cfsupp 8825  Basecbs 16475  +gcplusg 16557  Scalarcsca 16560   ·𝑠 cvsca 16561  0gc0g 16705   Σg cgsu 16706  Mndcmnd 17902  Grpcgrp 18035  invgcminusg 18036  CMndccmn 18828  LModclmod 19556  LSpanclspn 19665   freeLMod cfrlm 20806   LIndF clindf 20864 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-map 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-sup 8898  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12886  df-fzo 13027  df-seq 13363  df-hash 13684  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-gsum 16708  df-prds 16713  df-pws 16715  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-mhm 17946  df-submnd 17947  df-grp 18038  df-minusg 18039  df-sbg 18040  df-mulg 18157  df-subg 18208  df-ghm 18288  df-cntz 18379  df-cmn 18830  df-abl 18831  df-mgp 19162  df-ur 19174  df-ring 19221  df-subrg 19455  df-lmod 19558  df-lss 19626  df-lsp 19666  df-lmhm 19716  df-lbs 19769  df-sra 19866  df-rgmod 19867  df-nzr 19952  df-dsmm 20792  df-frlm 20807  df-uvc 20843  df-lindf 20866 This theorem is referenced by:  islindf5  20899  islinds5  30846  fedgmul  30913  matunitlindflem1  34755  aacllem  44730
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