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| Description: Version of fsumcn 24895 for two-argument mappings. (Contributed by Mario Carneiro, 6-May-2014.) | 
| Ref | Expression | 
|---|---|
| fsumcn.3 | ⊢ 𝐾 = (TopOpen‘ℂfld) | 
| fsumcn.4 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | 
| fsumcn.5 | ⊢ (𝜑 → 𝐴 ∈ Fin) | 
| fsum2cn.7 | ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌)) | 
| fsum2cn.8 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾)) | 
| Ref | Expression | 
|---|---|
| fsum2cn | ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑢Σ𝑘 ∈ 𝐴 𝐵 | |
| 2 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑣Σ𝑘 ∈ 𝐴 𝐵 | |
| 3 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
| 4 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥𝑣 | |
| 5 | nfcsb1v 3922 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵 | |
| 6 | 4, 5 | nfcsbw 3924 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵 | 
| 7 | 3, 6 | nfsum 15728 | . . . 4 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵 | 
| 8 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
| 9 | nfcsb1v 3922 | . . . . 5 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵 | |
| 10 | 8, 9 | nfsum 15728 | . . . 4 ⊢ Ⅎ𝑦Σ𝑘 ∈ 𝐴 ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵 | 
| 11 | csbeq1a 3912 | . . . . . 6 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵) | |
| 12 | csbeq1a 3912 | . . . . . 6 ⊢ (𝑦 = 𝑣 → ⦋𝑢 / 𝑥⦌𝐵 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | |
| 13 | 11, 12 | sylan9eq 2796 | . . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐵 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | 
| 14 | 13 | sumeq2sdv 15740 | . . . 4 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | 
| 15 | 1, 2, 7, 10, 14 | cbvmpo 7528 | . . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ Σ𝑘 ∈ 𝐴 𝐵) = (𝑢 ∈ 𝑋, 𝑣 ∈ 𝑌 ↦ Σ𝑘 ∈ 𝐴 ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | 
| 16 | vex 3483 | . . . . . . . 8 ⊢ 𝑢 ∈ V | |
| 17 | vex 3483 | . . . . . . . 8 ⊢ 𝑣 ∈ V | |
| 18 | 16, 17 | op2ndd 8026 | . . . . . . 7 ⊢ (𝑧 = 〈𝑢, 𝑣〉 → (2nd ‘𝑧) = 𝑣) | 
| 19 | 18 | csbeq1d 3902 | . . . . . 6 ⊢ (𝑧 = 〈𝑢, 𝑣〉 → ⦋(2nd ‘𝑧) / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵 = ⦋𝑣 / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵) | 
| 20 | 16, 17 | op1std 8025 | . . . . . . . 8 ⊢ (𝑧 = 〈𝑢, 𝑣〉 → (1st ‘𝑧) = 𝑢) | 
| 21 | 20 | csbeq1d 3902 | . . . . . . 7 ⊢ (𝑧 = 〈𝑢, 𝑣〉 → ⦋(1st ‘𝑧) / 𝑥⦌𝐵 = ⦋𝑢 / 𝑥⦌𝐵) | 
| 22 | 21 | csbeq2dv 3905 | . . . . . 6 ⊢ (𝑧 = 〈𝑢, 𝑣〉 → ⦋𝑣 / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | 
| 23 | 19, 22 | eqtrd 2776 | . . . . 5 ⊢ (𝑧 = 〈𝑢, 𝑣〉 → ⦋(2nd ‘𝑧) / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵 = ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | 
| 24 | 23 | sumeq2sdv 15740 | . . . 4 ⊢ (𝑧 = 〈𝑢, 𝑣〉 → Σ𝑘 ∈ 𝐴 ⦋(2nd ‘𝑧) / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵 = Σ𝑘 ∈ 𝐴 ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | 
| 25 | 24 | mpompt 7548 | . . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ Σ𝑘 ∈ 𝐴 ⦋(2nd ‘𝑧) / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵) = (𝑢 ∈ 𝑋, 𝑣 ∈ 𝑌 ↦ Σ𝑘 ∈ 𝐴 ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | 
| 26 | 15, 25 | eqtr4i 2767 | . 2 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ Σ𝑘 ∈ 𝐴 𝐵) = (𝑧 ∈ (𝑋 × 𝑌) ↦ Σ𝑘 ∈ 𝐴 ⦋(2nd ‘𝑧) / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵) | 
| 27 | fsumcn.3 | . . 3 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
| 28 | fsumcn.4 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
| 29 | fsum2cn.7 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌)) | |
| 30 | txtopon 23600 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌))) | |
| 31 | 28, 29, 30 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐽 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌))) | 
| 32 | fsumcn.5 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
| 33 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑢𝐵 | |
| 34 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑣𝐵 | |
| 35 | 33, 34, 6, 9, 13 | cbvmpo 7528 | . . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑢 ∈ 𝑋, 𝑣 ∈ 𝑌 ↦ ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | 
| 36 | 23 | mpompt 7548 | . . . . 5 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ⦋(2nd ‘𝑧) / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵) = (𝑢 ∈ 𝑋, 𝑣 ∈ 𝑌 ↦ ⦋𝑣 / 𝑦⦌⦋𝑢 / 𝑥⦌𝐵) | 
| 37 | 35, 36 | eqtr4i 2767 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ⦋(2nd ‘𝑧) / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵) | 
| 38 | fsum2cn.8 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾)) | |
| 39 | 37, 38 | eqeltrrid 2845 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ⦋(2nd ‘𝑧) / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾)) | 
| 40 | 27, 31, 32, 39 | fsumcn 24895 | . 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ Σ𝑘 ∈ 𝐴 ⦋(2nd ‘𝑧) / 𝑦⦌⦋(1st ‘𝑧) / 𝑥⦌𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾)) | 
| 41 | 26, 40 | eqeltrid 2844 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⦋csb 3898 〈cop 4631 ↦ cmpt 5224 × cxp 5682 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 1st c1st 8013 2nd c2nd 8014 Fincfn 8986 Σcsu 15723 TopOpenctopn 17467 ℂfldccnfld 21365 TopOnctopon 22917 Cn ccn 23233 ×t ctx 23569 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-icc 13395 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cn 23236 df-cnp 23237 df-tx 23571 df-hmeo 23764 df-xms 24331 df-ms 24332 df-tms 24333 | 
| This theorem is referenced by: dipcn 30740 | 
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