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Mirrors > Home > ILE Home > Th. List > hash2iun1dif1 | GIF version |
Description: The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.) |
Ref | Expression |
---|---|
hash2iun1dif1.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
hash2iun1dif1.b | ⊢ 𝐵 = (𝐴 ∖ {𝑥}) |
hash2iun1dif1.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin) |
hash2iun1dif1.da | ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) |
hash2iun1dif1.db | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶) |
hash2iun1dif1.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (♯‘𝐶) = 1) |
Ref | Expression |
---|---|
hash2iun1dif1 | ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hash2iun1dif1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | hash2iun1dif1.b | . . . 4 ⊢ 𝐵 = (𝐴 ∖ {𝑥}) | |
3 | 1 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ Fin) |
4 | snfig 6788 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ Fin) | |
5 | 4 | adantl 275 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ Fin) |
6 | snssi 3722 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
7 | 6 | adantl 275 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴) |
8 | diffifi 6868 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ {𝑥} ∈ Fin ∧ {𝑥} ⊆ 𝐴) → (𝐴 ∖ {𝑥}) ∈ Fin) | |
9 | 3, 5, 7, 8 | syl3anc 1233 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ∈ Fin) |
10 | 2, 9 | eqeltrid 2257 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) |
11 | hash2iun1dif1.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin) | |
12 | hash2iun1dif1.da | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) | |
13 | hash2iun1dif1.db | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶) | |
14 | 1, 10, 11, 12, 13 | hash2iun 11429 | . 2 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 (♯‘𝐶)) |
15 | hash2iun1dif1.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (♯‘𝐶) = 1) | |
16 | 15 | 2sumeq2dv 11321 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 (♯‘𝐶) = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 1) |
17 | 1cnd 7923 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 1 ∈ ℂ) | |
18 | fsumconst 11404 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑦 ∈ 𝐵 1 = ((♯‘𝐵) · 1)) | |
19 | 10, 17, 18 | syl2anc 409 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑦 ∈ 𝐵 1 = ((♯‘𝐵) · 1)) |
20 | 19 | sumeq2dv 11318 | . . 3 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 1 = Σ𝑥 ∈ 𝐴 ((♯‘𝐵) · 1)) |
21 | 2 | a1i 9 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = (𝐴 ∖ {𝑥})) |
22 | 21 | fveq2d 5498 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (♯‘𝐵) = (♯‘(𝐴 ∖ {𝑥}))) |
23 | hashdifsn 10741 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝑥})) = ((♯‘𝐴) − 1)) | |
24 | 1, 23 | sylan 281 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝑥})) = ((♯‘𝐴) − 1)) |
25 | 22, 24 | eqtrd 2203 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (♯‘𝐵) = ((♯‘𝐴) − 1)) |
26 | 25 | oveq1d 5865 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((♯‘𝐵) · 1) = (((♯‘𝐴) − 1) · 1)) |
27 | 26 | sumeq2dv 11318 | . . 3 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 ((♯‘𝐵) · 1) = Σ𝑥 ∈ 𝐴 (((♯‘𝐴) − 1) · 1)) |
28 | hashcl 10702 | . . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
29 | 1, 28 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
30 | 29 | nn0cnd 9177 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ) |
31 | peano2cnm 8172 | . . . . . . 7 ⊢ ((♯‘𝐴) ∈ ℂ → ((♯‘𝐴) − 1) ∈ ℂ) | |
32 | 30, 31 | syl 14 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℂ) |
33 | 32 | mulid1d 7924 | . . . . 5 ⊢ (𝜑 → (((♯‘𝐴) − 1) · 1) = ((♯‘𝐴) − 1)) |
34 | 33 | sumeq2ad 11319 | . . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 (((♯‘𝐴) − 1) · 1) = Σ𝑥 ∈ 𝐴 ((♯‘𝐴) − 1)) |
35 | fsumconst 11404 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ((♯‘𝐴) − 1) ∈ ℂ) → Σ𝑥 ∈ 𝐴 ((♯‘𝐴) − 1) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) | |
36 | 1, 32, 35 | syl2anc 409 | . . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 ((♯‘𝐴) − 1) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
37 | 34, 36 | eqtrd 2203 | . . 3 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 (((♯‘𝐴) − 1) · 1) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
38 | 20, 27, 37 | 3eqtrd 2207 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 1 = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
39 | 14, 16, 38 | 3eqtrd 2207 | 1 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ∖ cdif 3118 ⊆ wss 3121 {csn 3581 ∪ ciun 3871 Disj wdisj 3964 ‘cfv 5196 (class class class)co 5850 Fincfn 6714 ℂcc 7759 1c1 7762 · cmul 7766 − cmin 8077 ℕ0cn0 9122 ♯chash 10696 Σcsu 11303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 ax-caucvg 7881 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-disj 3965 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-frec 6367 df-1o 6392 df-oadd 6396 df-er 6509 df-en 6715 df-dom 6716 df-fin 6717 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-n0 9123 df-z 9200 df-uz 9475 df-q 9566 df-rp 9598 df-fz 9953 df-fzo 10086 df-seqfrec 10389 df-exp 10463 df-ihash 10697 df-cj 10793 df-re 10794 df-im 10795 df-rsqrt 10949 df-abs 10950 df-clim 11229 df-sumdc 11304 |
This theorem is referenced by: (None) |
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