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Theorem probun 33881

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 1211	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑃 ∈ Prob)
2		simplr 766	. . . 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 4453	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))
5	4	biimpi 215	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐴 = (𝐴 ∖ 𝐵))
6		difeq1 4115	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = (𝐵 ∖ 𝐵))
7		difid 4370	. . . . . . . . . . 11 ⊢ (𝐵 ∖ 𝐵) = ∅
8	6, 7	eqtrdi 2787	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅)
9	5, 8	sylan9eqr 2793	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 = ∅)
10		eqtr2 2755	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = ∅) → 𝐵 = ∅)
11	9, 10	syldan 590	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 = ∅)
12	9, 11	uneq12d 4164	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅))
13		unidm 4152	. . . . . . . 8 ⊢ (∅ ∪ ∅) = ∅
14	12, 13	eqtrdi 2787	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = ∅)
15	14	fveq2d 6895	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = (𝑃‘∅))
16		probnul 33876	. . . . . 6 ⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0)
17	15, 16	sylan9eqr 2793	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘(𝐴 ∪ 𝐵)) = 0)
18	9	fveq2d 6895	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) = (𝑃‘∅))
19	18, 16	sylan9eqr 2793	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘𝐴) = 0)
20	11	fveq2d 6895	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) = (𝑃‘∅))
21	20, 16	sylan9eqr 2793	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘𝐵) = 0)
22	19, 21	oveq12d 7430	. . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝑃‘𝐴) + (𝑃‘𝐵)) = (0 + 0))
23		00id 11396	. . . . . 6 ⊢ (0 + 0) = 0
24	22, 23	eqtrdi 2787	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝑃‘𝐴) + (𝑃‘𝐵)) = 0)
25	17, 24	eqtr4d 2774	. . . 4 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
26	1, 2, 3, 25	syl12anc 834	. . 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 623	. . . . . 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 1190	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑃 ∈ Prob)
33		prssi 4824	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → {𝐴, 𝐵} ⊆ dom 𝑃)
34	33	3ad2ant2 1133	. . . . . . . . 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 5432	. . . . . . . . 9 ⊢ {𝐴, 𝐵} ∈ V
37	36	elpw 4606	. . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ 𝒫 dom 𝑃 ↔ {𝐴, 𝐵} ⊆ dom 𝑃)
38	35, 37	sylibr 233	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝐴, 𝐵} ∈ 𝒫 dom 𝑃)
39		prct 32371	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → {𝐴, 𝐵} ≼ ω)
40	39	3ad2ant2 1133	. . . . . . . 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 1198	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ dom 𝑃)
43		simp2r 1199	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ dom 𝑃)
44		simp3 1137	. . . . . . . . 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 5144	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝑥 ↔ (𝐴 ∩ 𝐵) = ∅))
48	42, 43, 44, 47	syl3anc 1370	. . . . . . . 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 33880	. . . . . . 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 1373	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘∪ {𝐴, 𝐵}) = Σ^*𝑥 ∈ {𝐴, 𝐵} (𝑃‘𝑥))
52		uniprg 4925	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
53	52	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘∪ {𝐴, 𝐵}) = (𝑃‘(𝐴 ∪ 𝐵)))
54	53	3ad2ant2 1133	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘∪ {𝐴, 𝐵}) = (𝑃‘(𝐴 ∪ 𝐵)))
55		fveq2 6891	. . . . . . . . . 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 6891	. . . . . . . . . 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 33343	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]+∞)
60		simp1 1135	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝑃 ∈ Prob)
61		prob01 33875	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃) → (𝑃‘𝐴) ∈ (0[,]1))
62	60, 42, 61	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐴) ∈ (0[,]1))
63	59, 62	sselid 3980	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐴) ∈ (0[,]+∞))
64		prob01 33875	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘𝐵) ∈ (0[,]1))
65	60, 43, 64	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐵) ∈ (0[,]1))
66	59, 65	sselid 3980	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐵) ∈ (0[,]+∞))
67	56, 58, 42, 43, 63, 66, 44	esumpr 33527	. . . . . . . 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 2747	. . . . . . 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 231	. . . . 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 580	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)))
72		unitssre 13483	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
73		simpll1 1211	. . . . . . 7 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑃 ∈ Prob)
74		simpll2 1212	. . . . . . 7 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ dom 𝑃)
75	73, 74, 61	syl2anc 583	. . . . . 6 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) ∈ (0[,]1))
76	72, 75	sselid 3980	. . . . 5 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) ∈ ℝ)
77		simpll3 1213	. . . . . . 7 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ dom 𝑃)
78	73, 77, 64	syl2anc 583	. . . . . 6 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) ∈ (0[,]1))
79	72, 78	sselid 3980	. . . . 5 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) ∈ ℝ)
80		rexadd 13218	. . . . 5 ⊢ (((𝑃‘𝐴) ∈ ℝ ∧ (𝑃‘𝐵) ∈ ℝ) → ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
81	76, 79, 80	syl2anc 583	. . . 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 2771	. . 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 3028	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 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {cpr 4630 ∪ cuni 4908 Disj wdisj 5113 class class class wbr 5148 dom cdm 5676 ‘cfv 6543 (class class class)co 7412 ωcom 7859 ≼ cdom 8943 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 +∞cpnf 11252 +_𝑒 cxad 13097 [,]cicc 13334 Σ^*cesum 33488 Probcprb 33869

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-ac2 10464 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-acn 9943 df-ac 10117 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ioc 13336 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-fac 14241 df-bc 14270 df-hash 14298 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15640 df-ef 16018 df-sin 16020 df-cos 16021 df-pi 16023 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-ordt 17454 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-ps 18529 df-tsr 18530 df-plusf 18570 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-subrng 20442 df-subrg 20467 df-abv 20656 df-lmod 20704 df-scaf 20705 df-sra 21018 df-rgmod 21019 df-psmet 21224 df-xmet 21225 df-met 21226 df-bl 21227 df-mopn 21228 df-fbas 21229 df-fg 21230 df-cnfld 21233 df-top 22715 df-topon 22732 df-topsp 22754 df-bases 22768 df-cld 22842 df-ntr 22843 df-cls 22844 df-nei 22921 df-lp 22959 df-perf 22960 df-cn 23050 df-cnp 23051 df-haus 23138 df-tx 23385 df-hmeo 23578 df-fil 23669 df-fm 23761 df-flim 23762 df-flf 23763 df-tmd 23895 df-tgp 23896 df-tsms 23950 df-trg 23983 df-xms 24145 df-ms 24146 df-tms 24147 df-nm 24410 df-ngp 24411 df-nrg 24413 df-nlm 24414 df-ii 24716 df-cncf 24717 df-limc 25714 df-dv 25715 df-log 26404 df-esum 33489 df-siga 33570 df-meas 33657 df-prob 33870

This theorem is referenced by: probdif 33882

W3C validator