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Theorem mndpsuppss 41437
 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 511 . . . . . 6 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ (¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)))
2 nne 2794 . . . . . . 7 (¬ (𝐴𝑥) ≠ (0g𝑀) ↔ (𝐴𝑥) = (0g𝑀))
3 nne 2794 . . . . . . 7 (¬ (𝐵𝑥) ≠ (0g𝑀) ↔ (𝐵𝑥) = (0g𝑀))
42, 3anbi12i 732 . . . . . 6 ((¬ (𝐴𝑥) ≠ (0g𝑀) ∧ ¬ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
51, 4bitri 264 . . . . 5 (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) ↔ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)))
6 elmapfn 7824 . . . . . . . . . . . 12 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 Fn 𝑉)
76ad2antrl 763 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐴 Fn 𝑉)
87adantr 481 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐴 Fn 𝑉)
9 elmapfn 7824 . . . . . . . . . . . 12 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵 Fn 𝑉)
109ad2antll 764 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 Fn 𝑉)
1110adantr 481 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝐵 Fn 𝑉)
12 simplr 791 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝑉𝑋)
1312adantr 481 . . . . . . . . . 10 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → 𝑉𝑋)
14 inidm 3800 . . . . . . . . . 10 (𝑉𝑉) = 𝑉
15 simplrl 799 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐴𝑥) = (0g𝑀))
16 simplrr 800 . . . . . . . . . 10 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → (𝐵𝑥) = (0g𝑀))
178, 11, 13, 13, 14, 15, 16ofval 6859 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) ∧ 𝑥𝑉) → ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
1817an32s 845 . . . . . . . 8 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) = ((0g𝑀)(+g𝑀)(0g𝑀)))
19 eqid 2621 . . . . . . . . . . . 12 (Base‘𝑀) = (Base‘𝑀)
20 eqid 2621 . . . . . . . . . . . 12 (0g𝑀) = (0g𝑀)
2119, 20mndidcl 17229 . . . . . . . . . . 11 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
2221ancli 573 . . . . . . . . . 10 (𝑀 ∈ Mnd → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
2322ad4antr 767 . . . . . . . . 9 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → (𝑀 ∈ Mnd ∧ (0g𝑀) ∈ (Base‘𝑀)))
24 eqid 2621 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
2519, 24, 20mndlid 17232 . . . . . . . . 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 2655 . . . . . . 7 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
28 nne 2794 . . . . . . 7 (¬ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀) ↔ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) = (0g𝑀))
2927, 28sylibr 224 . . . . . 6 (((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) ∧ ((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀))) → ¬ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀))
3029ex 450 . . . . 5 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑥) = (0g𝑀) ∧ (𝐵𝑥) = (0g𝑀)) → ¬ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)))
315, 30syl5bi 232 . . . 4 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (¬ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀)) → ¬ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)))
3231con4d 114 . . 3 ((((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) ∧ 𝑥𝑉) → (((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀) → ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))))
3332ss2rabdv 3662 . 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 6861 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 (+g𝑀)𝐵) Fn 𝑉)
35 fnfun 5946 . . . . 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 6632 . . . . 5 (𝐴𝑓 (+g𝑀)𝐵) ∈ V
3837a1i 11 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 (+g𝑀)𝐵) ∈ V)
39 fvex 6158 . . . . 5 (0g𝑀) ∈ V
4039a1i 11 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (0g𝑀) ∈ V)
41 suppval1 7246 . . . 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)})
4236, 38, 40, 41syl3anc 1323 . . 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)})
4312, 7, 10offvalfv 41406 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 (+g𝑀)𝐵) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
4443dmeqd 5286 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → dom (𝐴𝑓 (+g𝑀)𝐵) = dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))))
45 ovex 6632 . . . . . 6 ((𝐴𝑣)(+g𝑀)(𝐵𝑣)) ∈ V
46 eqid 2621 . . . . . 6 (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣)))
4745, 46dmmpti 5980 . . . . 5 dom (𝑣𝑉 ↦ ((𝐴𝑣)(+g𝑀)(𝐵𝑣))) = 𝑉
4844, 47syl6eq 2671 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → dom (𝐴𝑓 (+g𝑀)𝐵) = 𝑉)
49 rabeq 3179 . . . 4 (dom (𝐴𝑓 (+g𝑀)𝐵) = 𝑉 → {𝑥 ∈ dom (𝐴𝑓 (+g𝑀)𝐵) ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
5048, 49syl 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)})
5142, 50eqtrd 2655 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) = {𝑥𝑉 ∣ ((𝐴𝑓 (+g𝑀)𝐵)‘𝑥) ≠ (0g𝑀)})
52 elmapfun 7825 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → Fun 𝐴)
53 id 22 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴 ∈ (𝑅𝑚 𝑉))
5439a1i 11 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → (0g𝑀) ∈ V)
55 suppval1 7246 . . . . . . 7 ((Fun 𝐴𝐴 ∈ (𝑅𝑚 𝑉) ∧ (0g𝑀) ∈ V) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
5652, 53, 54, 55syl3anc 1323 . . . . . 6 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)})
57 elmapi 7823 . . . . . . 7 (𝐴 ∈ (𝑅𝑚 𝑉) → 𝐴:𝑉𝑅)
58 fdm 6008 . . . . . . 7 (𝐴:𝑉𝑅 → dom 𝐴 = 𝑉)
59 rabeq 3179 . . . . . . 7 (dom 𝐴 = 𝑉 → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
6057, 58, 593syl 18 . . . . . 6 (𝐴 ∈ (𝑅𝑚 𝑉) → {𝑥 ∈ dom 𝐴 ∣ (𝐴𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
6156, 60eqtrd 2655 . . . . 5 (𝐴 ∈ (𝑅𝑚 𝑉) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
6261ad2antrl 763 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)})
63 elmapfun 7825 . . . . . . 7 (𝐵 ∈ (𝑅𝑚 𝑉) → Fun 𝐵)
6463ad2antll 764 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → Fun 𝐵)
65 simprr 795 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → 𝐵 ∈ (𝑅𝑚 𝑉))
66 suppval1 7246 . . . . . 6 ((Fun 𝐵𝐵 ∈ (𝑅𝑚 𝑉) ∧ (0g𝑀) ∈ V) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
6764, 65, 40, 66syl3anc 1323 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)})
68 elmapi 7823 . . . . . . . 8 (𝐵 ∈ (𝑅𝑚 𝑉) → 𝐵:𝑉𝑅)
69 fdm 6008 . . . . . . . 8 (𝐵:𝑉𝑅 → dom 𝐵 = 𝑉)
7068, 69syl 17 . . . . . . 7 (𝐵 ∈ (𝑅𝑚 𝑉) → dom 𝐵 = 𝑉)
7170ad2antll 764 . . . . . 6 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → dom 𝐵 = 𝑉)
72 rabeq 3179 . . . . . 6 (dom 𝐵 = 𝑉 → {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
7371, 72syl 17 . . . . 5 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → {𝑥 ∈ dom 𝐵 ∣ (𝐵𝑥) ≠ (0g𝑀)} = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
7467, 73eqtrd 2655 . . . 4 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐵 supp (0g𝑀)) = {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)})
7562, 74uneq12d 3746 . . 3 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}))
76 unrab 3874 . . 3 ({𝑥𝑉 ∣ (𝐴𝑥) ≠ (0g𝑀)} ∪ {𝑥𝑉 ∣ (𝐵𝑥) ≠ (0g𝑀)}) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))}
7775, 76syl6eq 2671 . 2 (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))) = {𝑥𝑉 ∣ ((𝐴𝑥) ≠ (0g𝑀) ∨ (𝐵𝑥) ≠ (0g𝑀))})
7833, 51, 773sstr4d 3627 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   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  {crab 2911  Vcvv 3186   ∪ cun 3553   ⊆ wss 3555   ↦ cmpt 4673  dom cdm 5074  Fun wfun 5841   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ∘𝑓 cof 6848   supp csupp 7240   ↑𝑚 cmap 7802  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Mndcmnd 17215 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-1st 7113  df-2nd 7114  df-supp 7241  df-map 7804  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216 This theorem is referenced by:  mndpsuppfi  41441
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