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Theorem probun 33233

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 1212	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑃 ∈ Prob)
2		simplr 767	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 = 𝐵)
3		simpr 485	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
4		disj3 4446	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))
5	4	biimpi 215	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐴 = (𝐴 ∖ 𝐵))
6		difeq1 4108	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = (𝐵 ∖ 𝐵))
7		difid 4363	. . . . . . . . . . 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 591	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 = ∅)
12	9, 11	uneq12d 4157	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅))
13		unidm 4145	. . . . . . . 8 ⊢ (∅ ∪ ∅) = ∅
14	12, 13	eqtrdi 2787	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = ∅)
15	14	fveq2d 6879	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = (𝑃‘∅))
16		probnul 33228	. . . . . 6 ⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0)
17	15, 16	sylan9eqr 2793	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘(𝐴 ∪ 𝐵)) = 0)
18	9	fveq2d 6879	. . . . . . . 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 6879	. . . . . . . 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 7408	. . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝑃‘𝐴) + (𝑃‘𝐵)) = (0 + 0))
23		00id 11368	. . . . . 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 835	. . 3 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
27	26	ex 413	. 2 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵))))
28		3anass 1095	. . . . . . 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 1089	. . . . . 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 277	. . . . 5 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ↔ (𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵))
32		simpl1 1191	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑃 ∈ Prob)
33		prssi 4814	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝐴, 𝐵} ⊆ dom 𝑃)
36		prex 5422	. . . . . . . . 9 ⊢ {𝐴, 𝐵} ∈ V
37	36	elpw 4597	. . . . . . . 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 31805	. . . . . . . . 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 481	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝐴, 𝐵} ≼ ω)
42		simp2l 1199	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ dom 𝑃)
43		simp2r 1200	. . . . . . . . 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 5134	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝑥 ↔ (𝐴 ∩ 𝐵) = ∅))
48	42, 43, 44, 47	syl3anc 1371	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝑥 ↔ (𝐴 ∩ 𝐵) = ∅))
49	48	biimpar 478	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → Disj 𝑥 ∈ {𝐴, 𝐵}𝑥)
50		probcun 33232	. . . . . . 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 1374	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘∪ {𝐴, 𝐵}) = Σ^*𝑥 ∈ {𝐴, 𝐵} (𝑃‘𝑥))
52		uniprg 4915	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
53	52	fveq2d 6879	. . . . . . . . 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 6875	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑃‘𝑥) = (𝑃‘𝐴))
56	55	adantl 482	. . . . . . . . 9 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 = 𝐴) → (𝑃‘𝑥) = (𝑃‘𝐴))
57		fveq2 6875	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑃‘𝑥) = (𝑃‘𝐵))
58	57	adantl 482	. . . . . . . . 9 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 = 𝐵) → (𝑃‘𝑥) = (𝑃‘𝐵))
59		unitssxrge0 32695	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]+∞)
60		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝑃 ∈ Prob)
61		prob01 33227	. . . . . . . . . . 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 3973	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐴) ∈ (0[,]+∞))
64		prob01 33227	. . . . . . . . . . 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 3973	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐵) ∈ (0[,]+∞))
67	56, 58, 42, 43, 63, 66, 44	esumpr 32879	. . . . . . . 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 481	. . . . . 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 581	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)))
72		unitssre 13455	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
73		simpll1 1212	. . . . . . 7 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑃 ∈ Prob)
74		simpll2 1213	. . . . . . 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 3973	. . . . 5 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) ∈ ℝ)
77		simpll3 1214	. . . . . . 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 3973	. . . . 5 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) ∈ ℝ)
80		rexadd 13190	. . . . 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 2771	. . 3 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
83	82	ex 413	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∖ cdif 3938 ∪ cun 3939 ∩ cin 3940 ⊆ wss 3941 ∅c0 4315 𝒫 cpw 4593 {cpr 4621 ∪ cuni 4898 Disj wdisj 5103 class class class wbr 5138 dom cdm 5666 ‘cfv 6529 (class class class)co 7390 ωcom 7835 ≼ cdom 8917 ℝcr 11088 0cc0 11089 1c1 11090 + caddc 11092 +∞cpnf 11224 +_𝑒 cxad 13069 [,]cicc 13306 Σ^*cesum 32840 Probcprb 33221

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-inf2 9615 ax-ac2 10437 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 ax-pre-sup 11167 ax-addf 11168 ax-mulf 11169

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-tp 4624 df-op 4626 df-uni 4899 df-int 4941 df-iun 4989 df-iin 4990 df-disj 5104 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-isom 6538 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7650 df-om 7836 df-1st 7954 df-2nd 7955 df-supp 8126 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-2o 8446 df-er 8683 df-map 8802 df-pm 8803 df-ixp 8872 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-fsupp 9342 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9484 df-dju 9875 df-card 9913 df-acn 9916 df-ac 10090 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-div 11851 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-7 12259 df-8 12260 df-9 12261 df-n0 12452 df-z 12538 df-dec 12657 df-uz 12802 df-q 12912 df-rp 12954 df-xneg 13071 df-xadd 13072 df-xmul 13073 df-ioo 13307 df-ioc 13308 df-ico 13309 df-icc 13310 df-fz 13464 df-fzo 13607 df-fl 13736 df-mod 13814 df-seq 13946 df-exp 14007 df-fac 14213 df-bc 14242 df-hash 14270 df-shft 14993 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15612 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-struct 17059 df-sets 17076 df-slot 17094 df-ndx 17106 df-base 17124 df-ress 17153 df-plusg 17189 df-mulr 17190 df-starv 17191 df-sca 17192 df-vsca 17193 df-ip 17194 df-tset 17195 df-ple 17196 df-ds 17198 df-unif 17199 df-hom 17200 df-cco 17201 df-rest 17347 df-topn 17348 df-0g 17366 df-gsum 17367 df-topgen 17368 df-pt 17369 df-prds 17372 df-ordt 17426 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-ps 18498 df-tsr 18499 df-plusf 18539 df-mgm 18540 df-sgrp 18589 df-mnd 18600 df-mhm 18644 df-submnd 18645 df-grp 18794 df-minusg 18795 df-sbg 18796 df-mulg 18920 df-subg 18972 df-cntz 19144 df-cmn 19611 df-abl 19612 df-mgp 19944 df-ur 19961 df-ring 20013 df-cring 20014 df-subrg 20305 df-abv 20369 df-lmod 20417 df-scaf 20418 df-sra 20729 df-rgmod 20730 df-psmet 20865 df-xmet 20866 df-met 20867 df-bl 20868 df-mopn 20869 df-fbas 20870 df-fg 20871 df-cnfld 20874 df-top 22320 df-topon 22337 df-topsp 22359 df-bases 22373 df-cld 22447 df-ntr 22448 df-cls 22449 df-nei 22526 df-lp 22564 df-perf 22565 df-cn 22655 df-cnp 22656 df-haus 22743 df-tx 22990 df-hmeo 23183 df-fil 23274 df-fm 23366 df-flim 23367 df-flf 23368 df-tmd 23500 df-tgp 23501 df-tsms 23555 df-trg 23588 df-xms 23750 df-ms 23751 df-tms 23752 df-nm 24015 df-ngp 24016 df-nrg 24018 df-nlm 24019 df-ii 24317 df-cncf 24318 df-limc 25307 df-dv 25308 df-log 25989 df-esum 32841 df-siga 32922 df-meas 33009 df-prob 33222

This theorem is referenced by: probdif 33234

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