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Theorem islindf4 20158
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 4315 . . . . 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 1098 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ LMod)
3 simprll 801 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑙 ∈ (Base‘𝑅))
4 ffvelrn 6343 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝐼𝐵𝑗𝐼) → (𝐹𝑗) ∈ 𝐵)
543ad2antl3 1223 . . . . . . . . . . . . . . . . 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 2620 . . . . . . . . . . . . . . . . 17 (invg𝑊) = (invg𝑊)
11 eqid 2620 . . . . . . . . . . . . . . . . 17 (invg𝑅) = (invg𝑅)
12 eqid 2620 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
137, 8, 9, 10, 11, 12lmodvsinv 19017 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘𝑅) ∧ (𝐹𝑗) ∈ 𝐵) → (((invg𝑅)‘𝑙) · (𝐹𝑗)) = ((invg𝑊)‘(𝑙 · (𝐹𝑗))))
142, 3, 6, 13syl3anc 1324 . . . . . . . . . . . . . . 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 2622 . . . . . . . . . . . . . 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 18851 . . . . . . . . . . . . . . . 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 18861 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘𝑅) ∧ (𝐹𝑗) ∈ 𝐵) → (𝑙 · (𝐹𝑗)) ∈ 𝐵)
192, 3, 6, 18syl3anc 1324 . . . . . . . . . . . . . . 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 18892 . . . . . . . . . . . . . . . . 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 1099 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐼𝑋)
24 difexg 4799 . . . . . . . . . . . . . . . . 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 802 . . . . . . . . . . . . . . . . . 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 7864 . . . . . . . . . . . . . . . . . 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 1100 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹:𝐼𝐵)
30 difss 3729 . . . . . . . . . . . . . . . . . 18 (𝐼 ∖ {𝑗}) ⊆ 𝐼
31 fssres 6057 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝐼𝐵 ∧ (𝐼 ∖ {𝑗}) ⊆ 𝐼) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
3229, 30, 31sylancl 693 . . . . . . . . . . . . . . . . 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 18883 . . . . . . . . . . . . . . . 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 795 . . . . . . . . . . . . . . . . 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 18884 . . . . . . . . . . . . . . . 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 18297 . . . . . . . . . . . . . . 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 2620 . . . . . . . . . . . . . . . 16 (+g𝑊) = (+g𝑊)
397, 38, 20, 10grpinvid2 17452 . . . . . . . . . . . . . . 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 1324 . . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑗𝐼)
42 fsnunf2 6437 . . . . . . . . . . . . . . . . . . 19 ((𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) ∧ 𝑗𝐼𝑙 ∈ (Base‘𝑅)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅))
4328, 41, 3, 42syl3anc 1324 . . . . . . . . . . . . . . . . . 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 18883 . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝑗𝐼)
46 simpl 473 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) → 𝑙 ∈ (Base‘𝑅))
4745, 46anim12i 589 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (𝑗𝐼𝑙 ∈ (Base‘𝑅)))
48 elmapfun 7866 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → Fun 𝑦)
49 fdm 6038 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → dom 𝑦 = (𝐼 ∖ {𝑗}))
50 neldifsnd 4313 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (dom 𝑦 = (𝐼 ∖ {𝑗}) → ¬ 𝑗 ∈ (𝐼 ∖ {𝑗}))
51 df-nel 2895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∉ dom 𝑦 ↔ ¬ 𝑗 ∈ dom 𝑦)
52 eleq2 2688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (dom 𝑦 = (𝐼 ∖ {𝑗}) → (𝑗 ∈ dom 𝑦𝑗 ∈ (𝐼 ∖ {𝑗})))
5352notbid 308 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝑦 = (𝐼 ∖ {𝑗}) → (¬ 𝑗 ∈ dom 𝑦 ↔ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})))
5451, 53syl5bb 272 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (dom 𝑦 = (𝐼 ∖ {𝑗}) → (𝑗 ∉ dom 𝑦 ↔ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})))
5550, 54mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (dom 𝑦 = (𝐼 ∖ {𝑗}) → 𝑗 ∉ dom 𝑦)
5649, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → 𝑗 ∉ dom 𝑦)
5727, 56syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → 𝑗 ∉ dom 𝑦)
5848, 57jca 554 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → (Fun 𝑦𝑗 ∉ dom 𝑦))
5958adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) → (Fun 𝑦𝑗 ∉ dom 𝑦))
6059adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (Fun 𝑦𝑗 ∉ dom 𝑦))
6147, 60jca 554 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → ((𝑗𝐼𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦𝑗 ∉ dom 𝑦)))
62 funsnfsupp 8284 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗𝐼𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦𝑗 ∉ dom 𝑦)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌𝑦 finSupp 𝑌))
6362bicomd 213 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗𝐼𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦𝑗 ∉ dom 𝑦)) → (𝑦 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
6461, 63syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (𝑦 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
6564biimpd 219 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (𝑦 finSupp 𝑌 → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
6665impr 648 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌)
678, 12, 9, 7, 2, 43, 29, 23, 20, 34, 66lcomfsupp 18884 . . . . . . . . . . . . . . . . 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 )
68 incom 3797 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ({𝑗} ∩ (𝐼 ∖ {𝑗}))
69 disjdif 4031 . . . . . . . . . . . . . . . . . . 19 ({𝑗} ∩ (𝐼 ∖ {𝑗})) = ∅
7068, 69eqtri 2642 . . . . . . . . . . . . . . . . . 18 ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ∅
7170a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ∅)
72 difsnid 4332 . . . . . . . . . . . . . . . . . . 19 (𝑗𝐼 → ((𝐼 ∖ {𝑗}) ∪ {𝑗}) = 𝐼)
7372eqcomd 2626 . . . . . . . . . . . . . . . . . 18 (𝑗𝐼𝐼 = ((𝐼 ∖ {𝑗}) ∪ {𝑗}))
7441, 73syl 17 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐼 = ((𝐼 ∖ {𝑗}) ∪ {𝑗}))
757, 20, 38, 22, 23, 44, 67, 71, 74gsumsplit 18309 . . . . . . . . . . . . . . . 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}))))
76 vex 3198 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
77 snex 4899 . . . . . . . . . . . . . . . . . . . . 21 {⟨𝑗, 𝑙⟩} ∈ V
7876, 77unex 6941 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∈ V
79 simpl3 1064 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐹:𝐼𝐵)
80 simpl2 1063 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐼𝑋)
81 fex 6475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:𝐼𝐵𝐼𝑋) → 𝐹 ∈ V)
8279, 80, 81syl2anc 692 . . . . . . . . . . . . . . . . . . . . 21 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐹 ∈ V)
8382adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹 ∈ V)
84 offres 7148 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∈ V ∧ 𝐹 ∈ V) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
8578, 83, 84sylancr 694 . . . . . . . . . . . . . . . . . . 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}))))
86 ffn 6032 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → 𝑦 Fn (𝐼 ∖ {𝑗}))
8728, 86syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 Fn (𝐼 ∖ {𝑗}))
88 neldifsn 4312 . . . . . . . . . . . . . . . . . . . . 21 ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})
89 fsnunres 6439 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 Fn (𝐼 ∖ {𝑗}) ∧ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) = 𝑦)
9087, 88, 89sylancl 693 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) = 𝑦)
9190oveq1d 6650 . . . . . . . . . . . . . . . . . . 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}))))
9285, 91eqtrd 2654 . . . . . . . . . . . . . . . . . 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}))))
9392oveq2d 6651 . . . . . . . . . . . . . . . . 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})))))
94 ffn 6032 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹):𝐼𝐵 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) Fn 𝐼)
9544, 94syl 17 . . . . . . . . . . . . . . . . . . . . 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)
96 fnressn 6410 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) Fn 𝐼𝑗𝐼) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}) = {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗)⟩})
9795, 41, 96syl2anc 692 . . . . . . . . . . . . . . . . . . . 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)⟩})
98 ffn 6032 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼)
9943, 98syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼)
100 ffn 6032 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹:𝐼𝐵𝐹 Fn 𝐼)
10129, 100syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹 Fn 𝐼)
102 fnfvof 6896 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼𝐹 Fn 𝐼) ∧ (𝐼𝑋𝑗𝐼)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹𝑗)))
10399, 101, 23, 41, 102syl22anc 1325 . . . . . . . . . . . . . . . . . . . . . . 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)))
104 fndm 5978 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 Fn (𝐼 ∖ {𝑗}) → dom 𝑦 = (𝐼 ∖ {𝑗}))
105104eleq2d 2685 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 Fn (𝐼 ∖ {𝑗}) → (𝑗 ∈ dom 𝑦𝑗 ∈ (𝐼 ∖ {𝑗})))
10688, 105mtbiri 317 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 Fn (𝐼 ∖ {𝑗}) → ¬ 𝑗 ∈ dom 𝑦)
107 vex 3198 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑗 ∈ V
108 vex 3198 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑙 ∈ V
109 fsnunfv 6438 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑗 ∈ V ∧ 𝑙 ∈ V ∧ ¬ 𝑗 ∈ dom 𝑦) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
110107, 108, 109mp3an12 1412 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑗 ∈ dom 𝑦 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
11187, 106, 1103syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
112111oveq1d 6650 . . . . . . . . . . . . . . . . . . . . . . 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)))
113103, 112eqtrd 2654 . . . . . . . . . . . . . . . . . . . . . 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)))
114113opeq2d 4400 . . . . . . . . . . . . . . . . . . . . 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))⟩)
115114sneqd 4180 . . . . . . . . . . . . . . . . . . . 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))⟩})
116 ovex 6663 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 · (𝐹𝑗)) ∈ V
117 fmptsn 6418 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ V ∧ (𝑙 · (𝐹𝑗)) ∈ V) → {⟨𝑗, (𝑙 · (𝐹𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗))))
118107, 116, 117mp2an 707 . . . . . . . . . . . . . . . . . . . . 21 {⟨𝑗, (𝑙 · (𝐹𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))
119118a1i 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))))
12097, 115, 1193eqtrd 2658 . . . . . . . . . . . . . . . . . . 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))))
121120oveq2d 6651 . . . . . . . . . . . . . . . . . 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)))))
122 cmnmnd 18189 . . . . . . . . . . . . . . . . . . . 20 (𝑊 ∈ CMnd → 𝑊 ∈ Mnd)
12322, 122syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ Mnd)
124107a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑗 ∈ V)
125 eqidd 2621 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑗 → (𝑙 · (𝐹𝑗)) = (𝑙 · (𝐹𝑗)))
1267, 125gsumsn 18335 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Mnd ∧ 𝑗 ∈ V ∧ (𝑙 · (𝐹𝑗)) ∈ 𝐵) → (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))) = (𝑙 · (𝐹𝑗)))
127123, 124, 19, 126syl3anc 1324 . . . . . . . . . . . . . . . . . 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)))
128121, 127eqtrd 2654 . . . . . . . . . . . . . . . . 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)))
12993, 128oveq12d 6653 . . . . . . . . . . . . . . . 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))))
13075, 129eqtr2d 2655 . . . . . . . . . . . . . . 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)))
131130eqeq1d 2622 . . . . . . . . . . . . . 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 ))
13215, 40, 1313bitrd 294 . . . . . . . . . . . . 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 ))
133111eqcomd 2626 . . . . . . . . . . . . . 14 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑙 = ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗))
134133eqeq1d 2622 . . . . . . . . . . . . 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))
135132, 134imbi12d 334 . . . . . . . . . . . 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)))
136135anassrs 679 . . . . . . . . . . 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)))
137136pm5.74da 722 . . . . . . . . . 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))))
138 impexp 462 . . . . . . . . . . 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)))
139138a1i 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))))
14064bicomd 213 . . . . . . . . . . 11 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌𝑦 finSupp 𝑌))
141140imbi1d 331 . . . . . . . . . 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))))
142137, 139, 1413bitr4d 300 . . . . . . . . 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))))
1431422ralbidva 2985 . . . . . . . 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))))
144 breq1 4647 . . . . . . . . . . 11 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
145 oveq1 6642 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥𝑓 · 𝐹) = ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹))
146145oveq2d 6651 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑊 Σg (𝑥𝑓 · 𝐹)) = (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)))
147146eqeq1d 2622 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 ))
148 fveq1 6177 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥𝑗) = ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗))
149148eqeq1d 2622 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑥𝑗) = 𝑌 ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))
150147, 149imbi12d 334 . . . . . . . . . . 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)))
151144, 150imbi12d 334 . . . . . . . . . 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))))
152151ralxpmap 7892 . . . . . . . . 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))))
153152adantl 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))))
154143, 153bitr4d 271 . . . . . . 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))))
155 breq1 4647 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 finSupp 𝑌𝑥 finSupp 𝑌))
156155ralrab 3362 . . . . . . 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)))
157154, 156syl6bbr 278 . . . . . 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)))
158 resima 5419 . . . . . . . . . . . . 13 ((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗})) = (𝐹 “ (𝐼 ∖ {𝑗}))
159158eqcomi 2629 . . . . . . . . . . . 12 (𝐹 “ (𝐼 ∖ {𝑗})) = ((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))
160159fveq2i 6181 . . . . . . . . . . 11 ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) = ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗})))
161160eleq2i 2691 . . . . . . . . . 10 ((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))))
162 eqid 2620 . . . . . . . . . . 11 (LSpan‘𝑊) = (LSpan‘𝑊)
16379, 30, 31sylancl 693 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
164 simpl1 1062 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝑊 ∈ LMod)
165243ad2ant2 1081 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐼 ∖ {𝑗}) ∈ V)
166165adantr 481 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (𝐼 ∖ {𝑗}) ∈ V)
167162, 7, 12, 8, 34, 9, 163, 164, 166ellspd 20122 . . . . . . . . . 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})))))))
168161, 167syl5bb 272 . . . . . . . . 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})))))))
169168imbi1d 331 . . . . . . . 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)))
170 r19.23v 3019 . . . . . . . 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))
171169, 170syl6bbr 278 . . . . . . 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)))
172171ralbidv 2983 . . . . . 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)))
173 fvex 6188 . . . . . . . . . . . 12 (Scalar‘𝑊) ∈ V
1748, 173eqeltri 2695 . . . . . . . . . . 11 𝑅 ∈ V
175 eqid 2620 . . . . . . . . . . . 12 (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
176 eqid 2620 . . . . . . . . . . . 12 {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌}
177175, 12, 34, 176frlmbas 20080 . . . . . . . . . . 11 ((𝑅 ∈ V ∧ 𝐼𝑋) → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
178174, 177mpan 705 . . . . . . . . . 10 (𝐼𝑋 → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
1791783ad2ant2 1081 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
180179adantr 481 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
181 islindf4.l . . . . . . . 8 𝐿 = (Base‘(𝑅 freeLMod 𝐼))
182180, 181syl6reqr 2673 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐿 = {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌})
183182raleqdv 3139 . . . . . 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)))
184157, 172, 1833bitr4d 300 . . . . 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)))
1851, 184syl5bb 272 . . . 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)))
1868lmodfgrp 18853 . . . . . . . 8 (𝑊 ∈ LMod → 𝑅 ∈ Grp)
18712, 34, 11grpinvnzcl 17468 . . . . . . . 8 ((𝑅 ∈ Grp ∧ 𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑙) ∈ ((Base‘𝑅) ∖ {𝑌}))
188186, 187sylan 488 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑙) ∈ ((Base‘𝑅) ∖ {𝑌}))
18912, 34, 11grpinvnzcl 17468 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}))
190186, 189sylan 488 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}))
191 eldifi 3724 . . . . . . . . . 10 (𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) → 𝑘 ∈ (Base‘𝑅))
19212, 11grpinvinv 17463 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝑘 ∈ (Base‘𝑅)) → ((invg𝑅)‘((invg𝑅)‘𝑘)) = 𝑘)
193186, 191, 192syl2an 494 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘((invg𝑅)‘𝑘)) = 𝑘)
194193eqcomd 2626 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → 𝑘 = ((invg𝑅)‘((invg𝑅)‘𝑘)))
195 fveq2 6178 . . . . . . . . . 10 (𝑙 = ((invg𝑅)‘𝑘) → ((invg𝑅)‘𝑙) = ((invg𝑅)‘((invg𝑅)‘𝑘)))
196195eqeq2d 2630 . . . . . . . . 9 (𝑙 = ((invg𝑅)‘𝑘) → (𝑘 = ((invg𝑅)‘𝑙) ↔ 𝑘 = ((invg𝑅)‘((invg𝑅)‘𝑘))))
197196rspcev 3304 . . . . . . . 8 ((((invg𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}) ∧ 𝑘 = ((invg𝑅)‘((invg𝑅)‘𝑘))) → ∃𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})𝑘 = ((invg𝑅)‘𝑙))
198190, 194, 197syl2anc 692 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ∃𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})𝑘 = ((invg𝑅)‘𝑙))
199 oveq1 6642 . . . . . . . . . 10 (𝑘 = ((invg𝑅)‘𝑙) → (𝑘 · (𝐹𝑗)) = (((invg𝑅)‘𝑙) · (𝐹𝑗)))
200199eleq1d 2684 . . . . . . . . 9 (𝑘 = ((invg𝑅)‘𝑙) → ((𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
201200notbid 308 . . . . . . . 8 (𝑘 = ((invg𝑅)‘𝑙) → (¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
202201adantl 482 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑘 = ((invg𝑅)‘𝑙)) → (¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
203188, 198, 202ralxfrd 4870 . . . . . 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})))))
2042033ad2ant1 1080 . . . . 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})))))
205204adantr 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})))))
206 simplr 791 . . . . . . . 8 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → 𝑗𝐼)
207 fvex 6188 . . . . . . . . . 10 (0g𝑅) ∈ V
20834, 207eqeltri 2695 . . . . . . . . 9 𝑌 ∈ V
209208fvconst2 6454 . . . . . . . 8 (𝑗𝐼 → ((𝐼 × {𝑌})‘𝑗) = 𝑌)
210206, 209syl 17 . . . . . . 7 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → ((𝐼 × {𝑌})‘𝑗) = 𝑌)
211210eqeq2d 2630 . . . . . 6 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → ((𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗) ↔ (𝑥𝑗) = 𝑌))
212211imbi2d 330 . . . . 5 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → (((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
213212ralbidva 2982 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
214185, 205, 2133bitr4d 300 . . 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))))
215214ralbidva 2982 . 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))))
2167, 9, 162, 8, 12, 34islindf2 20134 . 2 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑗𝐼𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
217175, 12, 181frlmbasf 20085 . . . . . . . 8 ((𝐼𝑋𝑥𝐿) → 𝑥:𝐼⟶(Base‘𝑅))
2182173ad2antl2 1222 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → 𝑥:𝐼⟶(Base‘𝑅))
219 ffn 6032 . . . . . . 7 (𝑥:𝐼⟶(Base‘𝑅) → 𝑥 Fn 𝐼)
220218, 219syl 17 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → 𝑥 Fn 𝐼)
221 fnconstg 6080 . . . . . . 7 (𝑌 ∈ V → (𝐼 × {𝑌}) Fn 𝐼)
222208, 221ax-mp 5 . . . . . 6 (𝐼 × {𝑌}) Fn 𝐼
223 eqfnfv 6297 . . . . . 6 ((𝑥 Fn 𝐼 ∧ (𝐼 × {𝑌}) Fn 𝐼) → (𝑥 = (𝐼 × {𝑌}) ↔ ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
224220, 222, 223sylancl 693 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → (𝑥 = (𝐼 × {𝑌}) ↔ ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
225224imbi2d 330 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → (((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0𝑥 = (𝐼 × {𝑌})) ↔ ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
226225ralbidva 2982 . . 3 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0𝑥 = (𝐼 × {𝑌})) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
227 r19.21v 2957 . . . . 5 (∀𝑗𝐼 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
228227ralbii 2977 . . . 4 (∀𝑥𝐿𝑗𝐼 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
229 ralcom 3093 . . . 4 (∀𝑥𝐿𝑗𝐼 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
230228, 229bitr3i 266 . . 3 (∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
231226, 230syl6bb 276 . 2 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0𝑥 = (𝐼 × {𝑌})) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
232215, 216, 2313bitr4d 300 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 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wnel 2894  wral 2909  wrex 2910  {crab 2913  Vcvv 3195  cdif 3564  cun 3565  cin 3566  wss 3567  c0 3907  {csn 4168  cop 4174   class class class wbr 4644  cmpt 4720   × cxp 5102  dom cdm 5104  cres 5106  cima 5107  Fun wfun 5870   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  𝑓 cof 6880  𝑚 cmap 7842   finSupp cfsupp 8260  Basecbs 15838  +gcplusg 15922  Scalarcsca 15925   ·𝑠 cvsca 15926  0gc0g 16081   Σg cgsu 16082  Mndcmnd 17275  Grpcgrp 17403  invgcminusg 17404  CMndccmn 18174  LModclmod 18844  LSpanclspn 18952   freeLMod cfrlm 20071   LIndF clindf 20124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-sup 8333  df-oi 8400  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-fzo 12450  df-seq 12785  df-hash 13101  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-sca 15938  df-vsca 15939  df-ip 15940  df-tset 15941  df-ple 15942  df-ds 15945  df-hom 15947  df-cco 15948  df-0g 16083  df-gsum 16084  df-prds 16089  df-pws 16091  df-mre 16227  df-mrc 16228  df-acs 16230  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-mhm 17316  df-submnd 17317  df-grp 17406  df-minusg 17407  df-sbg 17408  df-mulg 17522  df-subg 17572  df-ghm 17639  df-cntz 17731  df-cmn 18176  df-abl 18177  df-mgp 18471  df-ur 18483  df-ring 18530  df-subrg 18759  df-lmod 18846  df-lss 18914  df-lsp 18953  df-lmhm 19003  df-lbs 19056  df-sra 19153  df-rgmod 19154  df-nzr 19239  df-dsmm 20057  df-frlm 20072  df-uvc 20103  df-lindf 20126
This theorem is referenced by:  islindf5  20159  matunitlindflem1  33376  aacllem  42312
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