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Theorem mndpsuppss 42753
Description: The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
Hypothesis
Ref Expression
mndpsuppss.r 𝑅 = (Base‘𝑀)
Assertion
Ref Expression
mndpsuppss (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) ⊆ ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))))

Proof of Theorem mndpsuppss
Dummy variables 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioran 1006 . . . . . 6 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ (¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)))
2 nne 2940 . . . . . . 7 (¬ (𝐴𝑥) ≠ (0g𝑀) ↔ (𝐴𝑥) = (0g𝑀))
3 nne 2940 . . . . . . 7 (¬ (𝐵𝑥) ≠ (0g𝑀) ↔ (𝐵𝑥) = (0g𝑀))
42, 3anbi12i 620 . . . . . 6 ((¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
51, 4bitri 266 . . . . 5 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
6 elmapfn 8082 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 Fn 𝑉)
76ad2antrl 719 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐴 Fn 𝑉)
87adantr 472 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐴 Fn 𝑉)
9 elmapfn 8082 . . . . . . . . . . . 12 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵 Fn 𝑉)
109ad2antll 720 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 Fn 𝑉)
1110adantr 472 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐵 Fn 𝑉)
12 simplr 785 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝑉𝑋)
1312adantr 472 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝑉𝑋)
14 inidm 3981 . . . . . . . . . 10 (𝑉𝑉) = 𝑉
15 simplrl 795 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐴𝑥) = (0g𝑀))
16 simplrr 796 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐵𝑥) = (0g𝑀))
178, 11, 13, 13, 14, 15, 16ofval 7103 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
1817an32s 642 . . . . . . . 8 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
19 eqid 2764 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
20 eqid 2764 . . . . . . . . . . . 12 (0g𝑀) = (0g𝑀)
2119, 20mndidcl 17575 . . . . . . . . . . 11 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
2221ancli 544 . . . . . . . . . 10 (𝑀 ∈ Mnd → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
2322ad4antr 724 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
24 eqid 2764 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
2519, 24, 20mndlid 17578 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)) → ((0g𝑀)(+g𝑀)(0g𝑀)) = (0g𝑀))
2623, 25syl 17 . . . . . . . 8 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((0g𝑀)(+g𝑀)(0g𝑀)) = (0g𝑀))
2718, 26eqtrd 2798 . . . . . . 7 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
28 nne 2940 . . . . . . 7 (¬ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀) ↔ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
2927, 28sylibr 225 . . . . . 6 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ¬ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀))
3029ex 401 . . . . 5 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)) → ¬ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)))
315, 30syl5bi 233 . . . 4 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) → ¬ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)))
3231con4d 115 . . 3 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀) → ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))))
3332ss2rabdv 3842 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → {𝑥𝑉 ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)} ⊆ {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))})
347, 10, 12, 12, 14offn 7105 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 (+g𝑀)𝐵) Fn 𝑉)
35 fnfun 6165 . . . . 5 ((𝐴𝑓 (+g𝑀)𝐵) Fn 𝑉 → Fun (𝐴𝑓 (+g𝑀)𝐵))
3634, 35syl 17 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → Fun (𝐴𝑓 (+g𝑀)𝐵))
37 ovex 6873 . . . . 5 (𝐴𝑓 (+g𝑀)𝐵) ∈ V
3837a1i 11 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 (+g𝑀)𝐵) ∈ V)
39 fvexd 6389 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (0g𝑀) ∈ V)
40 suppval1 7502 . . . 4 ((Fun (𝐴𝑓 (+g𝑀)𝐵) ∧ (𝐴𝑓 (+g𝑀)𝐵) ∈ V ∧ (0g𝑀) ∈ V) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥 ∈ dom (𝐴𝑓 (+g𝑀)𝐵) ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
4136, 38, 39, 40syl3anc 1490 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥 ∈ dom (𝐴𝑓 (+g𝑀)𝐵) ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
4212, 7, 10offvalfv 42722 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 (+g𝑀)𝐵) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
4342dmeqd 5493 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → dom (𝐴𝑓 (+g𝑀)𝐵) = dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
44 ovex 6873 . . . . . 6 ((𝐴𝑣)(+g𝑀)(𝐵𝑣)) ∈ V
45 eqid 2764 . . . . . 6 (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣)))
4644, 45dmmpti 6200 . . . . 5 dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = 𝑉
4743, 46syl6eq 2814 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → dom (𝐴𝑓 (+g𝑀)𝐵) = 𝑉)
48 rabeq 3340 . . . 4 (dom (𝐴𝑓 (+g𝑀)𝐵) = 𝑉 → {𝑥 ∈ dom (𝐴𝑓 (+g𝑀)𝐵) ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
4947, 48syl 17 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → {𝑥 ∈ dom (𝐴𝑓 (+g𝑀)𝐵) ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
5041, 49eqtrd 2798 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥𝑉 ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
51 elmapfun 8083 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → Fun 𝐴)
52 id 22 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 ∈ (𝑅𝑚 𝑉))
53 fvexd 6389 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → (0g𝑀) ∈ V)
54 suppval1 7502 . . . . . . 7 ((Fun 𝐴𝐴 ∈ (𝑅𝑚 𝑉) ∧ (0g𝑀) ∈ V) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5551, 52, 53, 54syl3anc 1490 . . . . . 6 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
56 elmapi 8081 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
57 fdm 6230 . . . . . . 7 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
58 rabeq 3340 . . . . . . 7 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5956, 57, 583syl 18 . . . . . 6 (𝐴 ∈ (𝑅𝑚 𝑉) → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
6055, 59eqtrd 2798 . . . . 5 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
6160ad2antrl 719 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
62 elmapfun 8083 . . . . . . 7 (𝐵 ∈ (𝑅𝑚 𝑉) → Fun 𝐵)
6362ad2antll 720 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → Fun 𝐵)
64 simprr 789 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 ∈ (𝑅𝑚 𝑉))
65 suppval1 7502 . . . . . 6 ((Fun 𝐵𝐵 ∈ (𝑅𝑚 𝑉) ∧ (0g𝑀) ∈ V) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6663, 64, 39, 65syl3anc 1490 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
67 elmapi 8081 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵:𝑉𝑅)
6867fdmd 6231 . . . . . . 7 (𝐵 ∈ (𝑅𝑚 𝑉) → dom 𝐵 = 𝑉)
6968ad2antll 720 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → dom 𝐵 = 𝑉)
70 rabeq 3340 . . . . . 6 (dom 𝐵 = 𝑉 → {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
7169, 70syl 17 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
7266, 71eqtrd 2798 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
7361, 72uneq12d 3929 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}))
74 unrab 4061 . . 3 ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))}
7573, 74syl6eq 2814 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))})
7633, 50, 753sstr4d 3807 1 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) ⊆ ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873   = wceq 1652  wcel 2155  wne 2936  {crab 3058  Vcvv 3349  cun 3729  wss 3731  cmpt 4887  dom cdm 5276  Fun wfun 6061   Fn wfn 6062  wf 6063  cfv 6067  (class class class)co 6841  𝑓 cof 7092   supp csupp 7496  𝑚 cmap 8059  Basecbs 16131  +gcplusg 16215  0gc0g 16367  Mndcmnd 17561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-of 7094  df-1st 7365  df-2nd 7366  df-supp 7497  df-map 8061  df-0g 16369  df-mgm 17509  df-sgrp 17551  df-mnd 17562
This theorem is referenced by:  mndpsuppfi  42757
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