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Theorem hashunx 14389

Description: The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 14385. (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 14385	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
2	1	3expa 1127	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
3		hashcl 14359	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
4	3	nn0red 12533	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ)
5		hashcl 14359	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
6	5	nn0red 12533	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℝ)
7	4, 6	anim12i 621	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ))
8	7	adantr 483	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ))
9		rexadd 13225	. . . . . . 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 2762	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
12	2, 11	eqtrd 2791	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
13	12	expcom 416	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
14	13	3ad2ant3 1144	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
15		unexg 7715	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
16		unfir 9241	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
17	16	con3i 154	. . . . . 6 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴 ∪ 𝐵) ∈ Fin)
18		hashinf 14338	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ V ∧ ¬ (𝐴 ∪ 𝐵) ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = +∞)
19	15, 17, 18	syl2anr 605	. . . . 5 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘(𝐴 ∪ 𝐵)) = +∞)
20		ianor 992	. . . . . . 7 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ↔ (¬ 𝐴 ∈ Fin ∨ ¬ 𝐵 ∈ Fin))
21		simprl 778	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
22		simprr 780	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐵 ∈ 𝑊)
23		hashnfinnn0 14364	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) ∉ ℕ₀)
24	23	ex 415	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → (♯‘𝐴) ∉ ℕ₀))
25	24	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐴 ∈ Fin → (♯‘𝐴) ∉ ℕ₀))
26	25	impcom 410	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘𝐴) ∉ ℕ₀)
27		hashinfxadd 14388	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ₀) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = +∞)
28	21, 22, 26, 27	syl3anc 1386	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = +∞)
29	28	eqcomd 2762	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
30	29	ex 415	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
31		hashxrcl 14360	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ^*)
32		hashxrcl 14360	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑊 → (♯‘𝐵) ∈ ℝ^*)
33	31, 32	anim12i 621	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((♯‘𝐴) ∈ ℝ^* ∧ (♯‘𝐵) ∈ ℝ^*))
34	33	adantl 484	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐴) ∈ ℝ^* ∧ (♯‘𝐵) ∈ ℝ^*))
35		xaddcom 13233	. . . . . . . . . . . 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 780	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐵 ∈ 𝑊)
38		simprl 778	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
39		hashnfinnn0 14364	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑊 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) ∉ ℕ₀)
40	39	ex 415	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑊 → (¬ 𝐵 ∈ Fin → (♯‘𝐵) ∉ ℕ₀))
41	40	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ Fin → (♯‘𝐵) ∉ ℕ₀))
42	41	impcom 410	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘𝐵) ∉ ℕ₀)
43		hashinfxadd 14388	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ (♯‘𝐵) ∉ ℕ₀) → ((♯‘𝐵) +_𝑒 (♯‘𝐴)) = +∞)
44	37, 38, 42, 43	syl3anc 1386	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐵) +_𝑒 (♯‘𝐴)) = +∞)
45	36, 44	eqtrd 2791	. . . . . . . . . 10 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((♯‘𝐴) +_𝑒 (♯‘𝐵)) = +∞)
46	45	eqcomd 2762	. . . . . . . . 9 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
47	46	ex 415	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
48	30, 47	jaoi 866	. . . . . . 7 ⊢ ((¬ 𝐴 ∈ Fin ∨ ¬ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
49	20, 48	sylbi 219	. . . . . 6 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
50	49	imp 409	. . . . 5 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
51	19, 50	eqtrd 2791	. . . 4 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵)))
52	51	expcom 416	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
53	52	3adant3 1141	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ∪ 𝐵)) = ((♯‘𝐴) +_𝑒 (♯‘𝐵))))
54	14, 53	pm2.61d 180	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 398 ∨ wo 856 ∧ w3a 1095 = wceq 1554 ∈ wcel 2136 ∉ wnel 3055 Vcvv 3448 ∪ cun 3897 ∩ cin 3898 ∅c0 4280 ‘cfv 6510 (class class class)co 7385 Fincfn 8916 ℝcr 11062 + caddc 11066 +∞cpnf 11203 ℝ^*cxr 11205 ℕ₀cn0 12471 +_𝑒 cxad 13102 ♯chash 14333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-om 7836 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-oadd 8429 df-er 8666 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-dju 9849 df-card 9887 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-nn 12201 df-n0 12472 df-xnn0 12545 df-z 12559 df-uz 12830 df-xadd 13105 df-hash 14334

This theorem is referenced by: hashunsngx 14396 vtxdun 29621 vtxdginducedm1 29683 dimkerim 33878

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