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Theorem probun 31969

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 769	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 = 𝐵)
3		simpr 488	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
4		disj3 4353	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵))
5	4	biimpi 219	. . . . . . . . . 10 ⊢ ((𝐴 ∩ 𝐵) = ∅ → 𝐴 = (𝐴 ∖ 𝐵))
6		difeq1 4016	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = (𝐵 ∖ 𝐵))
7		difid 4269	. . . . . . . . . . 11 ⊢ (𝐵 ∖ 𝐵) = ∅
8	6, 7	eqtrdi 2790	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐴 ∖ 𝐵) = ∅)
9	5, 8	sylan9eqr 2796	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 = ∅)
10		eqtr2 2760	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = ∅) → 𝐵 = ∅)
11	9, 10	syldan 594	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 = ∅)
12	9, 11	uneq12d 4064	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = (∅ ∪ ∅))
13		unidm 4052	. . . . . . . 8 ⊢ (∅ ∪ ∅) = ∅
14	12, 13	eqtrdi 2790	. . . . . . 7 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) = ∅)
15	14	fveq2d 6691	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = (𝑃‘∅))
16		probnul 31964	. . . . . 6 ⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0)
17	15, 16	sylan9eqr 2796	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘(𝐴 ∪ 𝐵)) = 0)
18	9	fveq2d 6691	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) = (𝑃‘∅))
19	18, 16	sylan9eqr 2796	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘𝐴) = 0)
20	11	fveq2d 6691	. . . . . . . 8 ⊢ ((𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) = (𝑃‘∅))
21	20, 16	sylan9eqr 2796	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → (𝑃‘𝐵) = 0)
22	19, 21	oveq12d 7201	. . . . . 6 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝑃‘𝐴) + (𝑃‘𝐵)) = (0 + 0))
23		00id 10906	. . . . . 6 ⊢ (0 + 0) = 0
24	22, 23	eqtrdi 2790	. . . . 5 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 = 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅)) → ((𝑃‘𝐴) + (𝑃‘𝐵)) = 0)
25	17, 24	eqtr4d 2777	. . . 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 416	. 2 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 = 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵))))
28		3anass 1096	. . . . . . 7 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ↔ (𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃)))
29	28	anbi1i 627	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ↔ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃)) ∧ 𝐴 ≠ 𝐵))
30		df-3an 1090	. . . . . 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 281	. . . . 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 4719	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → {𝐴, 𝐵} ⊆ dom 𝑃)
34	33	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ⊆ dom 𝑃)
35	34	adantr 484	. . . . . . . 8 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝐴, 𝐵} ⊆ dom 𝑃)
36		prex 5309	. . . . . . . . 9 ⊢ {𝐴, 𝐵} ∈ V
37	36	elpw 4502	. . . . . . . 8 ⊢ ({𝐴, 𝐵} ∈ 𝒫 dom 𝑃 ↔ {𝐴, 𝐵} ⊆ dom 𝑃)
38	35, 37	sylibr 237	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → {𝐴, 𝐵} ∈ 𝒫 dom 𝑃)
39		prct 30637	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → {𝐴, 𝐵} ≼ ω)
40	39	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≼ ω)
41	40	adantr 484	. . . . . . 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 1139	. . . . . . . . 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 5036	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝑥 ↔ (𝐴 ∩ 𝐵) = ∅))
48	42, 43, 44, 47	syl3anc 1372	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝑥 ↔ (𝐴 ∩ 𝐵) = ∅))
49	48	biimpar 481	. . . . . . 7 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → Disj 𝑥 ∈ {𝐴, 𝐵}𝑥)
50		probcun 31968	. . . . . . 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 1375	. . . . . 6 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘∪ {𝐴, 𝐵}) = Σ^*𝑥 ∈ {𝐴, 𝐵} (𝑃‘𝑥))
52		uniprg 4823	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
53	52	fveq2d 6691	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘∪ {𝐴, 𝐵}) = (𝑃‘(𝐴 ∪ 𝐵)))
54	53	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘∪ {𝐴, 𝐵}) = (𝑃‘(𝐴 ∪ 𝐵)))
55		fveq2 6687	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑃‘𝑥) = (𝑃‘𝐴))
56	55	adantl 485	. . . . . . . . 9 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 = 𝐴) → (𝑃‘𝑥) = (𝑃‘𝐴))
57		fveq2 6687	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑃‘𝑥) = (𝑃‘𝐵))
58	57	adantl 485	. . . . . . . . 9 ⊢ (((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ 𝑥 = 𝐵) → (𝑃‘𝑥) = (𝑃‘𝐵))
59		unitssxrge0 31435	. . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]+∞)
60		simp1 1137	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → 𝑃 ∈ Prob)
61		prob01 31963	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃) → (𝑃‘𝐴) ∈ (0[,]1))
62	60, 42, 61	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐴) ∈ (0[,]1))
63	59, 62	sseldi 3885	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐴) ∈ (0[,]+∞))
64		prob01 31963	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Prob ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘𝐵) ∈ (0[,]1))
65	60, 43, 64	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐵) ∈ (0[,]1))
66	59, 65	sseldi 3885	. . . . . . . . 9 ⊢ ((𝑃 ∈ Prob ∧ (𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) → (𝑃‘𝐵) ∈ (0[,]+∞))
67	56, 58, 42, 43, 63, 66, 44	esumpr 31617	. . . . . . . 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 2755	. . . . . . 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 484	. . . . . 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 235	. . . . 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 584	. . . 4 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)))
72		unitssre 12986	. . . . . 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 587	. . . . . 6 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐴) ∈ (0[,]1))
76	72, 75	sseldi 3885	. . . . 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 587	. . . . . 6 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) ∈ (0[,]1))
79	72, 78	sseldi 3885	. . . . 5 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘𝐵) ∈ ℝ)
80		rexadd 12721	. . . . 5 ⊢ (((𝑃‘𝐴) ∈ ℝ ∧ (𝑃‘𝐵) ∈ ℝ) → ((𝑃‘𝐴) +_𝑒 (𝑃‘𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
81	76, 79, 80	syl2anc 587	. . . 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 2774	. . 3 ⊢ ((((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ≠ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵)))
83	82	ex 416	. 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 3022	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 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∖ cdif 3850 ∪ cun 3851 ∩ cin 3852 ⊆ wss 3853 ∅c0 4221 𝒫 cpw 4498 {cpr 4528 ∪ cuni 4806 Disj wdisj 5005 class class class wbr 5040 dom cdm 5535 ‘cfv 6350 (class class class)co 7183 ωcom 7612 ≼ cdom 8566 ℝcr 10627 0cc0 10628 1c1 10629 + caddc 10631 +∞cpnf 10763 +_𝑒 cxad 12601 [,]cicc 12837 Σ^*cesum 31578 Probcprb 31957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-inf2 9190 ax-ac2 9976 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 ax-addf 10707 ax-mulf 10708

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-disj 5006 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-of 7438 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-2o 8145 df-er 8333 df-map 8452 df-pm 8453 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-fi 8961 df-sup 8992 df-inf 8993 df-oi 9060 df-dju 9416 df-card 9454 df-acn 9457 df-ac 9629 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-z 12076 df-dec 12193 df-uz 12338 df-q 12444 df-rp 12486 df-xneg 12603 df-xadd 12604 df-xmul 12605 df-ioo 12838 df-ioc 12839 df-ico 12840 df-icc 12841 df-fz 12995 df-fzo 13138 df-fl 13266 df-mod 13342 df-seq 13474 df-exp 13535 df-fac 13739 df-bc 13768 df-hash 13796 df-shft 14529 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-limsup 14931 df-clim 14948 df-rlim 14949 df-sum 15149 df-ef 15526 df-sin 15528 df-cos 15529 df-pi 15531 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-starv 16696 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-unif 16704 df-hom 16705 df-cco 16706 df-rest 16812 df-topn 16813 df-0g 16831 df-gsum 16832 df-topgen 16833 df-pt 16834 df-prds 16837 df-ordt 16890 df-xrs 16891 df-qtop 16896 df-imas 16897 df-xps 16899 df-mre 16973 df-mrc 16974 df-acs 16976 df-ps 17939 df-tsr 17940 df-plusf 17980 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-mhm 18085 df-submnd 18086 df-grp 18235 df-minusg 18236 df-sbg 18237 df-mulg 18356 df-subg 18407 df-cntz 18578 df-cmn 19039 df-abl 19040 df-mgp 19372 df-ur 19384 df-ring 19431 df-cring 19432 df-subrg 19665 df-abv 19720 df-lmod 19768 df-scaf 19769 df-sra 20076 df-rgmod 20077 df-psmet 20222 df-xmet 20223 df-met 20224 df-bl 20225 df-mopn 20226 df-fbas 20227 df-fg 20228 df-cnfld 20231 df-top 21658 df-topon 21675 df-topsp 21697 df-bases 21710 df-cld 21783 df-ntr 21784 df-cls 21785 df-nei 21862 df-lp 21900 df-perf 21901 df-cn 21991 df-cnp 21992 df-haus 22079 df-tx 22326 df-hmeo 22519 df-fil 22610 df-fm 22702 df-flim 22703 df-flf 22704 df-tmd 22836 df-tgp 22837 df-tsms 22891 df-trg 22924 df-xms 23086 df-ms 23087 df-tms 23088 df-nm 23348 df-ngp 23349 df-nrg 23351 df-nlm 23352 df-ii 23642 df-cncf 23643 df-limc 24631 df-dv 24632 df-log 25313 df-esum 31579 df-siga 31660 df-meas 31747 df-prob 31958

This theorem is referenced by: probdif 31970

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