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Theorem hashunx 14342

Description: The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 14338. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

**Assertion**
Ref	Expression
hashunx	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))

**Proof of Theorem hashunx**
Step	Hyp	Ref	Expression
1		hashun 14338	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
2	1	3expa 1118	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
3		hashcl 14312	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
4	3	nn0red 12529	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ)
5		hashcl 14312	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
6	5	nn0red 12529	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℝ)
7	4, 6	anim12i 613	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ))
8	7	adantr 481	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ))
9		rexadd 13207	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
10	8, 9	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
11	10	eqcomd 2738	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
12	2, 11	eqtrd 2772	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
13	12	expcom 414	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
14	13	3ad2ant3 1135	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
15		unexg 7732	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
16		unfir 9310	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
17	16	con3i 154	. . . . . 6 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴 ∪ 𝐵) ∈ Fin)
18		hashinf 14291	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ V ∧ ¬ (𝐴 ∪ 𝐵) ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = +∞)
19	15, 17, 18	syl2anr 597	. . . . 5 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘(𝐴 ∪ 𝐵)) = +∞)
20		ianor 980	. . . . . . 7 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ↔ (¬ 𝐴 ∈ Fin ∨ ¬ 𝐵 ∈ Fin))
21		simprl 769	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
22		simprr 771	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐵 ∈ 𝑊)
23		hashnfinnn0 14317	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) ∉ ℕ₀)
24	23	ex 413	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → (♯‘𝐴) ∉ ℕ₀))
25	24	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐴 ∈ Fin → (♯‘𝐴) ∉ ℕ₀))
26	25	impcom 408	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘𝐴) ∉ ℕ₀)
27		hashinfxadd 14341	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ₀) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = +∞)
28	21, 22, 26, 27	syl3anc 1371	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = +∞)
29	28	eqcomd 2738	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
30	29	ex 413	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
31		hashxrcl 14313	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ^*)
32		hashxrcl 14313	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑊 → (♯‘𝐵) ∈ ℝ^*)
33	31, 32	anim12i 613	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((♯‘𝐴) ∈ ℝ^* ∧ (♯‘𝐵) ∈ ℝ^*))
34	33	adantl 482	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐴) ∈ ℝ^* ∧ (♯‘𝐵) ∈ ℝ^*))
35		xaddcom 13215	. . . . . . . . . . . 12 ⊢ (((♯‘𝐴) ∈ ℝ^* ∧ (♯‘𝐵) ∈ ℝ^*) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = ((♯‘𝐵) +_𝑒 (♯‘𝐴)))
36	34, 35	syl 17	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = ((♯‘𝐵) +_𝑒 (♯‘𝐴)))
37		simprr 771	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐵 ∈ 𝑊)
38		simprl 769	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
39		hashnfinnn0 14317	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑊 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) ∉ ℕ₀)
40	39	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑊 → (¬ 𝐵 ∈ Fin → (♯‘𝐵) ∉ ℕ₀))
41	40	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ Fin → (♯‘𝐵) ∉ ℕ₀))
42	41	impcom 408	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘𝐵) ∉ ℕ₀)
43		hashinfxadd 14341	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ (♯‘𝐵) ∉ ℕ₀) → ((♯‘𝐵) +_𝑒 (♯‘𝐴)) = +∞)
44	37, 38, 42, 43	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐵) +_𝑒 (♯‘𝐴)) = +∞)
45	36, 44	eqtrd 2772	. . . . . . . . . 10 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = +∞)
46	45	eqcomd 2738	. . . . . . . . 9 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
47	46	ex 413	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
48	30, 47	jaoi 855	. . . . . . 7 ⊢ ((¬ 𝐴 ∈ Fin ∨ ¬ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
49	20, 48	sylbi 216	. . . . . 6 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
50	49	imp 407	. . . . 5 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
51	19, 50	eqtrd 2772	. . . 4 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
52	51	expcom 414	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
53	52	3adant3 1132	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
54	14, 53	pm2.61d 179	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∉ wnel 3046 Vcvv 3474 ∪ cun 3945 ∩ cin 3946 ∅c0 4321 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℝcr 11105 + caddc 11109 +∞cpnf 11241 ℝ^*cxr 11243 ℕ₀cn0 12468 +_𝑒 cxad 13086 ♯chash 14286

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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-xadd 13089 df-hash 14287

This theorem is referenced by: hashunsngx 14349 vtxdun 28727 vtxdginducedm1 28789 dimkerim 32700

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