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Theorem probun 34422

Description: The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)

**Assertion**
Ref	Expression
probun	⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((𝐴 ∩ 𝐵) = ∅ → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵))))

Proof of Theorem probun Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll1 1213	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑃 ∈ Prob)
2		simplr 768	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 = 𝐵)
3		simpr 484	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
4		disj3 4402	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))
5	4	biimpi 216	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐴 = (𝐴 ∖ 𝐵))
6		difeq1 4067	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = (𝐵 ∖ 𝐵))
7		difid 4324	. . . . . . . . . . 11 ⊢ (𝐵 ∖ 𝐵) = ∅
8	6, 7	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅)
9	5, 8	sylan9eqr 2787	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 = ∅)
10		eqtr2 2751	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = ∅) → 𝐵 = ∅)
11	9, 10	syldan 591	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 = ∅)
12	9, 11	uneq12d 4117	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅))
13		unidm 4105	. . . . . . . 8 ⊢ (∅ ∪ ∅) = ∅
14	12, 13	eqtrdi 2781	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = ∅)
15	14	fveq2d 6821	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = (𝑃‘∅))
16		probnul 34417	. . . . . 6 ⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0)
17	15, 16	sylan9eqr 2787	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘(𝐴 ∪ 𝐵)) = 0)
18	9	fveq2d 6821	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) = (𝑃‘∅))
19	18, 16	sylan9eqr 2787	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘𝐴) = 0)
20	11	fveq2d 6821	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) = (𝑃‘∅))
21	20, 16	sylan9eqr 2787	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘𝐵) = 0)
22	19, 21	oveq12d 7359	. . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝑃‘𝐴) + (𝑃‘𝐵)) = (0 + 0))
23		00id 11280	. . . . . 6 ⊢ (0 + 0) = 0
24	22, 23	eqtrdi 2781	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝑃‘𝐴) + (𝑃‘𝐵)) = 0)
25	17, 24	eqtr4d 2768	. . . 4 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
26	1, 2, 3, 25	syl12anc 836	. . 3 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
27	26	ex 412	. 2 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵))))
28		3anass 1094	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ↔ (𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃)))
29	28	anbi1i 624	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ↔ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃)) ∧ 𝐴 ≠ 𝐵))
30		df-3an 1088	. . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ↔ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃)) ∧ 𝐴 ≠ 𝐵))
31	29, 30	bitr4i 278	. . . . 5 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ↔ (𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵))
32		simpl1 1192	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑃 ∈ Prob)
33		prssi 4771	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → {𝐴, 𝐵} ⊆ dom 𝑃)
34	33	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ⊆ dom 𝑃)
35	34	adantr 480	. . . . . . . 8 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝐴, 𝐵} ⊆ dom 𝑃)
36		prex 5373	. . . . . . . . 9 ⊢ {𝐴, 𝐵} ∈ V
37	36	elpw 4552	. . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ 𝒫 dom 𝑃 ↔ {𝐴, 𝐵} ⊆ dom 𝑃)
38	35, 37	sylibr 234	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝐴, 𝐵} ∈ 𝒫 dom 𝑃)
39		prct 32686	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → {𝐴, 𝐵} ≼ ω)
40	39	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≼ ω)
41	40	adantr 480	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝐴, 𝐵} ≼ ω)
42		simp2l 1200	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ dom 𝑃)
43		simp2r 1201	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ dom 𝑃)
44		simp3 1138	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵)
45		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
46		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
47	45, 46	disjprg 5085	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝑥 ↔ (𝐴 ∩ 𝐵) = ∅))
48	42, 43, 44, 47	syl3anc 1373	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝑥 ↔ (𝐴 ∩ 𝐵) = ∅))
49	48	biimpar 477	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → Disj 𝑥 ∈ {𝐴, 𝐵}𝑥)
50		probcun 34421	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ {𝐴, 𝐵} ∈ 𝒫 dom 𝑃 ∧ ({𝐴, 𝐵} ≼ ω ∧ Disj 𝑥 ∈ {𝐴, 𝐵}𝑥)) → (𝑃‘∪ {𝐴, 𝐵}) = Σ^*𝑥 ∈ {𝐴, 𝐵} (𝑃‘𝑥))
51	32, 38, 41, 49, 50	syl112anc 1376	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘∪ {𝐴, 𝐵}) = Σ^*𝑥 ∈ {𝐴, 𝐵} (𝑃‘𝑥))
52		uniprg 4873	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
53	52	fveq2d 6821	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘∪ {𝐴, 𝐵}) = (𝑃‘(𝐴 ∪ 𝐵)))
54	53	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘∪ {𝐴, 𝐵}) = (𝑃‘(𝐴 ∪ 𝐵)))
55		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑃‘𝑥) = (𝑃‘𝐴))
56	55	adantl 481	. . . . . . . . 9 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 = 𝐴) → (𝑃‘𝑥) = (𝑃‘𝐴))
57		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑃‘𝑥) = (𝑃‘𝐵))
58	57	adantl 481	. . . . . . . . 9 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 = 𝐵) → (𝑃‘𝑥) = (𝑃‘𝐵))
59		unitssxrge0 33903	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]+∞)
60		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝑃 ∈ Prob)
61		prob01 34416	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃) → (𝑃‘𝐴) ∈ (0[,]1))
62	60, 42, 61	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐴) ∈ (0[,]1))
63	59, 62	sselid 3930	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐴) ∈ (0[,]+∞))
64		prob01 34416	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘𝐵) ∈ (0[,]1))
65	60, 43, 64	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐵) ∈ (0[,]1))
66	59, 65	sselid 3930	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐵) ∈ (0[,]+∞))
67	56, 58, 42, 43, 63, 66, 44	esumpr 34069	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → Σ^*𝑥 ∈ {𝐴, 𝐵} (𝑃‘𝑥) = ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)))
68	54, 67	eqeq12d 2746	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → ((𝑃‘∪ {𝐴, 𝐵}) = Σ^*𝑥 ∈ {𝐴, 𝐵} (𝑃‘𝑥) ↔ (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵))))
69	68	adantr 480	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑃‘∪ {𝐴, 𝐵}) = Σ^*𝑥 ∈ {𝐴, 𝐵} (𝑃‘𝑥) ↔ (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵))))
70	51, 69	mpbid 232	. . . . 5 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)))
71	31, 70	sylanb 581	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)))
72		unitssre 13391	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
73		simpll1 1213	. . . . . . 7 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑃 ∈ Prob)
74		simpll2 1214	. . . . . . 7 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ dom 𝑃)
75	73, 74, 61	syl2anc 584	. . . . . 6 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) ∈ (0[,]1))
76	72, 75	sselid 3930	. . . . 5 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) ∈ ℝ)
77		simpll3 1215	. . . . . . 7 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ dom 𝑃)
78	73, 77, 64	syl2anc 584	. . . . . 6 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) ∈ (0[,]1))
79	72, 78	sselid 3930	. . . . 5 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) ∈ ℝ)
80		rexadd 13123	. . . . 5 ⊢ (((𝑃‘𝐴) ∈ ℝ ∧ (𝑃‘𝐵) ∈ ℝ) → ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
81	76, 79, 80	syl2anc 584	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
82	71, 81	eqtrd 2765	. . 3 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
83	82	ex 412	. 2 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵))))
84	27, 83	pm2.61dane 3013	1 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((𝐴 ∩ 𝐵) = ∅ → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2110 ≠ wne 2926 ∖ cdif 3897 ∪ cun 3898 ∩ cin 3899 ⊆ wss 3900 ∅c0 4281 𝒫 cpw 4548 {cpr 4576 ∪ cuni 4857 Disj wdisj 5056 class class class wbr 5089 dom cdm 5614 ‘cfv 6477 (class class class)co 7341 ωcom 7791 ≼ cdom 8862 ℝcr 10997 0cc0 10998 1c1 10999 + caddc 11001 +∞cpnf 11135 +_𝑒 cxad 13001 [,]cicc 13240 Σ^*cesum 34030 Probcprb 34410

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-ac2 10346 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077 ax-mulf 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9786 df-card 9824 df-acn 9827 df-ac 9999 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-fz 13400 df-fzo 13547 df-fl 13688 df-mod 13766 df-seq 13901 df-exp 13961 df-fac 14173 df-bc 14202 df-hash 14230 df-shft 14966 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-limsup 15370 df-clim 15387 df-rlim 15388 df-sum 15586 df-ef 15966 df-sin 15968 df-cos 15969 df-pi 15971 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-hom 17177 df-cco 17178 df-rest 17318 df-topn 17319 df-0g 17337 df-gsum 17338 df-topgen 17339 df-pt 17340 df-prds 17343 df-ordt 17397 df-xrs 17398 df-qtop 17403 df-imas 17404 df-xps 17406 df-mre 17480 df-mrc 17481 df-acs 17483 df-ps 18464 df-tsr 18465 df-plusf 18539 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-submnd 18684 df-grp 18841 df-minusg 18842 df-sbg 18843 df-mulg 18973 df-subg 19028 df-cntz 19222 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-cring 20147 df-subrng 20454 df-subrg 20478 df-abv 20717 df-lmod 20788 df-scaf 20789 df-sra 21100 df-rgmod 21101 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22802 df-topon 22819 df-topsp 22841 df-bases 22854 df-cld 22927 df-ntr 22928 df-cls 22929 df-nei 23006 df-lp 23044 df-perf 23045 df-cn 23135 df-cnp 23136 df-haus 23223 df-tx 23470 df-hmeo 23663 df-fil 23754 df-fm 23846 df-flim 23847 df-flf 23848 df-tmd 23980 df-tgp 23981 df-tsms 24035 df-trg 24068 df-xms 24228 df-ms 24229 df-tms 24230 df-nm 24490 df-ngp 24491 df-nrg 24493 df-nlm 24494 df-ii 24790 df-cncf 24791 df-limc 25787 df-dv 25788 df-log 26485 df-esum 34031 df-siga 34112 df-meas 34199 df-prob 34411

This theorem is referenced by: probdif 34423

W3C validator