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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 271 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ Fin) |
4 | snfig 6587 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ Fin) | |
5 | 4 | adantl 272 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑥} ∈ Fin) |
6 | snssi 3589 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
7 | 6 | adantl 272 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑥} ⊆ 𝐴) |
8 | diffifi 6666 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ {𝑥} ∈ Fin ∧ {𝑥} ⊆ 𝐴) → (𝐴 ∖ {𝑥}) ∈ Fin) | |
9 | 3, 5, 7, 8 | syl3anc 1175 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐴 ∖ {𝑥}) ∈ Fin) |
10 | 2, 9 | syl5eqel 2175 | . . 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 10936 | . 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 10823 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 (♯‘𝐶) = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 1) |
17 | 1cnd 7567 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 1 ∈ ℂ) | |
18 | fsumconst 10911 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑦 ∈ 𝐵 1 = ((♯‘𝐵) · 1)) | |
19 | 10, 17, 18 | syl2anc 404 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Σ𝑦 ∈ 𝐵 1 = ((♯‘𝐵) · 1)) |
20 | 19 | sumeq2dv 10820 | . . 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 5324 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (♯‘𝐵) = (♯‘(𝐴 ∖ {𝑥}))) |
23 | hashdifsn 10290 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝑥})) = ((♯‘𝐴) − 1)) | |
24 | 1, 23 | sylan 278 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝑥})) = ((♯‘𝐴) − 1)) |
25 | 22, 24 | eqtrd 2121 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (♯‘𝐵) = ((♯‘𝐴) − 1)) |
26 | 25 | oveq1d 5683 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((♯‘𝐵) · 1) = (((♯‘𝐴) − 1) · 1)) |
27 | 26 | sumeq2dv 10820 | . . 3 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 ((♯‘𝐵) · 1) = Σ𝑥 ∈ 𝐴 (((♯‘𝐴) − 1) · 1)) |
28 | hashcl 10252 | . . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
29 | 1, 28 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
30 | 29 | nn0cnd 8791 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ) |
31 | peano2cnm 7811 | . . . . . . 7 ⊢ ((♯‘𝐴) ∈ ℂ → ((♯‘𝐴) − 1) ∈ ℂ) | |
32 | 30, 31 | syl 14 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℂ) |
33 | 32 | mulid1d 7568 | . . . . 5 ⊢ (𝜑 → (((♯‘𝐴) − 1) · 1) = ((♯‘𝐴) − 1)) |
34 | 33 | sumeq2ad 10821 | . . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 (((♯‘𝐴) − 1) · 1) = Σ𝑥 ∈ 𝐴 ((♯‘𝐴) − 1)) |
35 | fsumconst 10911 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ((♯‘𝐴) − 1) ∈ ℂ) → Σ𝑥 ∈ 𝐴 ((♯‘𝐴) − 1) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) | |
36 | 1, 32, 35 | syl2anc 404 | . . . 4 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 ((♯‘𝐴) − 1) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
37 | 34, 36 | eqtrd 2121 | . . 3 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 (((♯‘𝐴) − 1) · 1) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
38 | 20, 27, 37 | 3eqtrd 2125 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 1 = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
39 | 14, 16, 38 | 3eqtrd 2125 | 1 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = ((♯‘𝐴) · ((♯‘𝐴) − 1))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 ∖ cdif 2999 ⊆ wss 3002 {csn 3452 ∪ ciun 3738 Disj wdisj 3830 ‘cfv 5030 (class class class)co 5668 Fincfn 6513 ℂcc 7411 1c1 7414 · cmul 7418 − cmin 7716 ℕ0cn0 8736 ♯chash 10246 Σcsu 10805 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-mulrcl 7507 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-precex 7518 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 ax-pre-mulgt0 7525 ax-pre-mulext 7526 ax-arch 7527 ax-caucvg 7528 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-if 3400 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-disj 3831 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-po 4134 df-iso 4135 df-iord 4204 df-on 4206 df-ilim 4207 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-isom 5039 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-recs 6086 df-irdg 6151 df-frec 6172 df-1o 6197 df-oadd 6201 df-er 6308 df-en 6514 df-dom 6515 df-fin 6516 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-reap 8115 df-ap 8122 df-div 8203 df-inn 8486 df-2 8544 df-3 8545 df-4 8546 df-n0 8737 df-z 8814 df-uz 9083 df-q 9168 df-rp 9198 df-fz 9488 df-fzo 9617 df-iseq 9916 df-seq3 9917 df-exp 10018 df-ihash 10247 df-cj 10339 df-re 10340 df-im 10341 df-rsqrt 10494 df-abs 10495 df-clim 10730 df-isum 10806 |
This theorem is referenced by: (None) |
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