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Theorem hashunx 14345

Description: The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 14341. (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 14341	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
2	1	3expa 1119	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
3		hashcl 14315	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
4	3	nn0red 12496	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ)
5		hashcl 14315	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
6	5	nn0red 12496	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℝ)
7	4, 6	anim12i 614	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ))
8	7	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ))
9		rexadd 13181	. . . . . . 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 2743	. . . . 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 413	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
14	13	3ad2ant3 1136	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
15		unexg 7694	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
16		unfir 9215	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
17	16	con3i 154	. . . . . 6 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴 ∪ 𝐵) ∈ Fin)
18		hashinf 14294	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ V ∧ ¬ (𝐴 ∪ 𝐵) ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = +∞)
19	15, 17, 18	syl2anr 598	. . . . 5 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘(𝐴 ∪ 𝐵)) = +∞)
20		ianor 984	. . . . . . 7 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ↔ (¬ 𝐴 ∈ Fin ∨ ¬ 𝐵 ∈ Fin))
21		simprl 771	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
22		simprr 773	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐵 ∈ 𝑊)
23		hashnfinnn0 14320	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) ∉ ℕ₀)
24	23	ex 412	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → (♯‘𝐴) ∉ ℕ₀))
25	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐴 ∈ Fin → (♯‘𝐴) ∉ ℕ₀))
26	25	impcom 407	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘𝐴) ∉ ℕ₀)
27		hashinfxadd 14344	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ₀) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = +∞)
28	21, 22, 26, 27	syl3anc 1374	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = +∞)
29	28	eqcomd 2743	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
30	29	ex 412	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
31		hashxrcl 14316	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ^*)
32		hashxrcl 14316	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑊 → (♯‘𝐵) ∈ ℝ^*)
33	31, 32	anim12i 614	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((♯‘𝐴) ∈ ℝ^* ∧ (♯‘𝐵) ∈ ℝ^*))
34	33	adantl 481	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐴) ∈ ℝ^* ∧ (♯‘𝐵) ∈ ℝ^*))
35		xaddcom 13189	. . . . . . . . . . . 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 773	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐵 ∈ 𝑊)
38		simprl 771	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
39		hashnfinnn0 14320	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑊 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) ∉ ℕ₀)
40	39	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑊 → (¬ 𝐵 ∈ Fin → (♯‘𝐵) ∉ ℕ₀))
41	40	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ Fin → (♯‘𝐵) ∉ ℕ₀))
42	41	impcom 407	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘𝐵) ∉ ℕ₀)
43		hashinfxadd 14344	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ (♯‘𝐵) ∉ ℕ₀) → ((♯‘𝐵) +_𝑒 (♯‘𝐴)) = +∞)
44	37, 38, 42, 43	syl3anc 1374	. . . . . . . . . . 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 2743	. . . . . . . . 9 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
47	46	ex 412	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
48	30, 47	jaoi 858	. . . . . . 7 ⊢ ((¬ 𝐴 ∈ Fin ∨ ¬ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
49	20, 48	sylbi 217	. . . . . 6 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
50	49	imp 406	. . . . 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 413	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
53	52	3adant3 1133	. 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 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∉ wnel 3037 Vcvv 3430 ∪ cun 3888 ∩ cin 3889 ∅c0 4274 ‘cfv 6496 (class class class)co 7364 Fincfn 8890 ℝcr 11034 + caddc 11038 +∞cpnf 11173 ℝ^*cxr 11175 ℕ₀cn0 12434 +_𝑒 cxad 13058 ♯chash 14289

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-oadd 8406 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9822 df-card 9860 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-nn 12172 df-n0 12435 df-xnn0 12508 df-z 12522 df-uz 12786 df-xadd 13061 df-hash 14290

This theorem is referenced by: hashunsngx 14352 vtxdun 29547 vtxdginducedm1 29609 dimkerim 33768

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