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Theorem hashunx 13175

Description: The size of the union of disjoint sets is the result of the extended real addition of their sizes, analogous to hashun 13171. (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 13171	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
2	1	3expa 1265	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
3		hashcl 13147	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ₀)
4	3	nn0red 11352	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℝ)
5		hashcl 13147	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ₀)
6	5	nn0red 11352	. . . . . . . . 9 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℝ)
7	4, 6	anim12i 590	. . . . . . . 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 12063	. . . . . . 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 2628	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((#‘𝐴) + (#‘𝐵)) = ((#‘𝐴) +_𝑒 (#‘𝐵)))
12	2, 11	eqtrd 2656	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) +_𝑒 (#‘𝐵)))
13	12	expcom 451	. . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) +_𝑒 (#‘𝐵))))
14	13	3ad2ant3 1084	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) +_𝑒 (#‘𝐵))))
15		unexg 6959	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
16		unfir 8228	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
17	16	con3i 150	. . . . . 6 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴 ∪ 𝐵) ∈ Fin)
18		hashinf 13122	. . . . . 6 ⊢ (((𝐴 ∪ 𝐵) ∈ V ∧ ¬ (𝐴 ∪ 𝐵) ∈ Fin) → (#‘(𝐴 ∪ 𝐵)) = +∞)
19	15, 17, 18	syl2anr 495	. . . . 5 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (#‘(𝐴 ∪ 𝐵)) = +∞)
20		ianor 509	. . . . . . 7 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ↔ (¬ 𝐴 ∈ Fin ∨ ¬ 𝐵 ∈ Fin))
21		simprl 794	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
22		simprr 796	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐵 ∈ 𝑊)
23		hashnfinnn0 13152	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘𝐴) ∉ ℕ₀)
24	23	ex 450	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ Fin → (#‘𝐴) ∉ ℕ₀))
25	24	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐴 ∈ Fin → (#‘𝐴) ∉ ℕ₀))
26	25	impcom 446	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (#‘𝐴) ∉ ℕ₀)
27		hashinfxadd 13174	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (#‘𝐴) ∉ ℕ₀) → ((#‘𝐴) +_𝑒 (#‘𝐵)) = +∞)
28	21, 22, 26, 27	syl3anc 1326	. . . . . . . . . 10 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((#‘𝐴) +_𝑒 (#‘𝐵)) = +∞)
29	28	eqcomd 2628	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((#‘𝐴) +_𝑒 (#‘𝐵)))
30	29	ex 450	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((#‘𝐴) +_𝑒 (#‘𝐵))))
31		hashxrcl 13148	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → (#‘𝐴) ∈ ℝ^*)
32		hashxrcl 13148	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑊 → (#‘𝐵) ∈ ℝ^*)
33	31, 32	anim12i 590	. . . . . . . . . . . . 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 12071	. . . . . . . . . . . 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 796	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐵 ∈ 𝑊)
38		simprl 794	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → 𝐴 ∈ 𝑉)
39		hashnfinnn0 13152	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑊 ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) ∉ ℕ₀)
40	39	ex 450	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑊 → (¬ 𝐵 ∈ Fin → (#‘𝐵) ∉ ℕ₀))
41	40	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝐵 ∈ Fin → (#‘𝐵) ∉ ℕ₀))
42	41	impcom 446	. . . . . . . . . . . 12 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (#‘𝐵) ∉ ℕ₀)
43		hashinfxadd 13174	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ (#‘𝐵) ∉ ℕ₀) → ((#‘𝐵) +_𝑒 (#‘𝐴)) = +∞)
44	37, 38, 42, 43	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((#‘𝐵) +_𝑒 (#‘𝐴)) = +∞)
45	36, 44	eqtrd 2656	. . . . . . . . . 10 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((#‘𝐴) +_𝑒 (#‘𝐵)) = +∞)
46	45	eqcomd 2628	. . . . . . . . 9 ⊢ ((¬ 𝐵 ∈ Fin ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((#‘𝐴) +_𝑒 (#‘𝐵)))
47	46	ex 450	. . . . . . . 8 ⊢ (¬ 𝐵 ∈ Fin → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((#‘𝐴) +_𝑒 (#‘𝐵))))
48	30, 47	jaoi 394	. . . . . . 7 ⊢ ((¬ 𝐴 ∈ Fin ∨ ¬ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((#‘𝐴) +_𝑒 (#‘𝐵))))
49	20, 48	sylbi 207	. . . . . 6 ⊢ (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → +∞ = ((#‘𝐴) +_𝑒 (#‘𝐵))))
50	49	imp 445	. . . . 5 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → +∞ = ((#‘𝐴) +_𝑒 (#‘𝐵)))
51	19, 50	eqtrd 2656	. . . 4 ⊢ ((¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) +_𝑒 (#‘𝐵)))
52	51	expcom 451	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) +_𝑒 (#‘𝐵))))
53	52	3adant3 1081	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (¬ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) +_𝑒 (#‘𝐵))))
54	14, 53	pm2.61d 170	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (#‘(𝐴 ∪ 𝐵)) = ((#‘𝐴) +_𝑒 (#‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℝcr 9935 + caddc 9939 +∞cpnf 10071 ℝ^*cxr 10073 ℕ₀cn0 11292 +_𝑒 cxad 11944 #chash 13117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-xadd 11947 df-hash 13118

This theorem is referenced by: vtxdun 26377 vtxdginducedm1 26439

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