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Theorem probun 34384

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 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 4477	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))
5	4	biimpi 216	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐴 = (𝐴 ∖ 𝐵))
6		difeq1 4142	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = (𝐵 ∖ 𝐵))
7		difid 4398	. . . . . . . . . . 11 ⊢ (𝐵 ∖ 𝐵) = ∅
8	6, 7	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅)
9	5, 8	sylan9eqr 2802	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 = ∅)
10		eqtr2 2764	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = ∅) → 𝐵 = ∅)
11	9, 10	syldan 590	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 = ∅)
12	9, 11	uneq12d 4192	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅))
13		unidm 4180	. . . . . . . 8 ⊢ (∅ ∪ ∅) = ∅
14	12, 13	eqtrdi 2796	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = ∅)
15	14	fveq2d 6924	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = (𝑃‘∅))
16		probnul 34379	. . . . . 6 ⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0)
17	15, 16	sylan9eqr 2802	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘(𝐴 ∪ 𝐵)) = 0)
18	9	fveq2d 6924	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) = (𝑃‘∅))
19	18, 16	sylan9eqr 2802	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘𝐴) = 0)
20	11	fveq2d 6924	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) = (𝑃‘∅))
21	20, 16	sylan9eqr 2802	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘𝐵) = 0)
22	19, 21	oveq12d 7466	. . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝑃‘𝐴) + (𝑃‘𝐵)) = (0 + 0))
23		00id 11465	. . . . . 6 ⊢ (0 + 0) = 0
24	22, 23	eqtrdi 2796	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝑃‘𝐴) + (𝑃‘𝐵)) = 0)
25	17, 24	eqtr4d 2783	. . . 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 1095	. . . . . . 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 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 278	. . . . 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 4846	. . . . . . . . . 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 5452	. . . . . . . . 9 ⊢ {𝐴, 𝐵} ∈ V
37	36	elpw 4626	. . . . . . . 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 32728	. . . . . . . . 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 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 5162	. . . . . . . . 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 477	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → Disj 𝑥 ∈ {𝐴, 𝐵}𝑥)
50		probcun 34383	. . . . . . 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 4947	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
53	52	fveq2d 6924	. . . . . . . . 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 6920	. . . . . . . . . 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 6920	. . . . . . . . . 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 33846	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]+∞)
60		simp1 1136	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝑃 ∈ Prob)
61		prob01 34378	. . . . . . . . . . 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 4006	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐴) ∈ (0[,]+∞))
64		prob01 34378	. . . . . . . . . . 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 4006	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐵) ∈ (0[,]+∞))
67	56, 58, 42, 43, 63, 66, 44	esumpr 34030	. . . . . . . 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 2756	. . . . . . 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 580	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)))
72		unitssre 13559	. . . . . 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 583	. . . . . 6 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) ∈ (0[,]1))
76	72, 75	sselid 4006	. . . . 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 583	. . . . . 6 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) ∈ (0[,]1))
79	72, 78	sselid 4006	. . . . 5 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) ∈ ℝ)
80		rexadd 13294	. . . . 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 2780	. . 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 3035	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 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∖ cdif 3973 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 𝒫 cpw 4622 {cpr 4650 ∪ cuni 4931 Disj wdisj 5133 class class class wbr 5166 dom cdm 5700 ‘cfv 6573 (class class class)co 7448 ωcom 7903 ≼ cdom 9001 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 +∞cpnf 11321 +_𝑒 cxad 13173 [,]cicc 13410 Σ^*cesum 33991 Probcprb 34372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-ac2 10532 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-ac 10185 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-ordt 17561 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-ps 18636 df-tsr 18637 df-plusf 18677 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-subrng 20572 df-subrg 20597 df-abv 20832 df-lmod 20882 df-scaf 20883 df-sra 21195 df-rgmod 21196 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-tmd 24101 df-tgp 24102 df-tsms 24156 df-trg 24189 df-xms 24351 df-ms 24352 df-tms 24353 df-nm 24616 df-ngp 24617 df-nrg 24619 df-nlm 24620 df-ii 24922 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-esum 33992 df-siga 34073 df-meas 34160 df-prob 34373

This theorem is referenced by: probdif 34385

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