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| Mirrors > Home > MPE Home > Th. List > psrascl | Structured version Visualization version GIF version | ||
| Description: Value of the scalar injection into the power series algebra. (Contributed by SN, 18-May-2025.) |
| Ref | Expression |
|---|---|
| psrascl.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
| psrascl.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
| psrascl.z | ⊢ 0 = (0g‘𝑅) |
| psrascl.k | ⊢ 𝐾 = (Base‘𝑅) |
| psrascl.a | ⊢ 𝐴 = (algSc‘𝑆) |
| psrascl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
| psrascl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
| psrascl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
| Ref | Expression |
|---|---|
| psrascl | ⊢ (𝜑 → (𝐴‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | psrascl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
| 2 | psrascl.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
| 3 | psrascl.s | . . . . . . 7 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
| 4 | psrascl.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
| 5 | psrascl.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 6 | 3, 4, 5 | psrsca 21967 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆)) |
| 7 | 6 | fveq2d 6910 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑆))) |
| 8 | 2, 7 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑆))) |
| 9 | 1, 8 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑆))) |
| 10 | psrascl.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑆) | |
| 11 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
| 12 | eqid 2737 | . . . 4 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆)) | |
| 13 | eqid 2737 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
| 14 | eqid 2737 | . . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
| 15 | 10, 11, 12, 13, 14 | asclval 21900 | . . 3 ⊢ (𝑋 ∈ (Base‘(Scalar‘𝑆)) → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑆)(1r‘𝑆))) |
| 16 | 9, 15 | syl 17 | . 2 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑋( ·𝑠 ‘𝑆)(1r‘𝑆))) |
| 17 | eqid 2737 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
| 18 | eqid 2737 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 19 | psrascl.d | . . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
| 20 | 3, 4, 5 | psrring 21990 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring) |
| 21 | 17, 14 | ringidcl 20262 | . . . 4 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆)) |
| 22 | 20, 21 | syl 17 | . . 3 ⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆)) |
| 23 | 3, 13, 2, 17, 18, 19, 1, 22 | psrvsca 21969 | . 2 ⊢ (𝜑 → (𝑋( ·𝑠 ‘𝑆)(1r‘𝑆)) = ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(1r‘𝑆))) |
| 24 | fnconstg 6796 | . . . . 5 ⊢ (𝑋 ∈ 𝐾 → (𝐷 × {𝑋}) Fn 𝐷) | |
| 25 | 1, 24 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐷 × {𝑋}) Fn 𝐷) |
| 26 | 3, 2, 19, 17, 22 | psrelbas 21954 | . . . . 5 ⊢ (𝜑 → (1r‘𝑆):𝐷⟶𝐾) |
| 27 | 26 | ffnd 6737 | . . . 4 ⊢ (𝜑 → (1r‘𝑆) Fn 𝐷) |
| 28 | ovexd 7466 | . . . . 5 ⊢ (𝜑 → (ℕ0 ↑m 𝐼) ∈ V) | |
| 29 | 19, 28 | rabexd 5340 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
| 30 | inidm 4227 | . . . 4 ⊢ (𝐷 ∩ 𝐷) = 𝐷 | |
| 31 | fvconst2g 7222 | . . . . 5 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑦 ∈ 𝐷) → ((𝐷 × {𝑋})‘𝑦) = 𝑋) | |
| 32 | 1, 31 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((𝐷 × {𝑋})‘𝑦) = 𝑋) |
| 33 | psrascl.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
| 34 | eqid 2737 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 35 | 3, 4, 5, 19, 33, 34, 14 | psr1 21991 | . . . . . . 7 ⊢ (𝜑 → (1r‘𝑆) = (𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 ))) |
| 36 | 35 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (1r‘𝑆) = (𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 ))) |
| 37 | 36 | fveq1d 6908 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((1r‘𝑆)‘𝑦) = ((𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 ))‘𝑦)) |
| 38 | eqeq1 2741 | . . . . . . . 8 ⊢ (𝑑 = 𝑦 → (𝑑 = (𝐼 × {0}) ↔ 𝑦 = (𝐼 × {0}))) | |
| 39 | 38 | ifbid 4549 | . . . . . . 7 ⊢ (𝑑 = 𝑦 → if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 ) = if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )) |
| 40 | eqid 2737 | . . . . . . 7 ⊢ (𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 )) = (𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 )) | |
| 41 | fvex 6919 | . . . . . . . 8 ⊢ (1r‘𝑅) ∈ V | |
| 42 | 33 | fvexi 6920 | . . . . . . . 8 ⊢ 0 ∈ V |
| 43 | 41, 42 | ifex 4576 | . . . . . . 7 ⊢ if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 ) ∈ V |
| 44 | 39, 40, 43 | fvmpt 7016 | . . . . . 6 ⊢ (𝑦 ∈ 𝐷 → ((𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 ))‘𝑦) = if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )) |
| 45 | 44 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 ))‘𝑦) = if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )) |
| 46 | 37, 45 | eqtrd 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((1r‘𝑆)‘𝑦) = if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )) |
| 47 | 25, 27, 29, 29, 30, 32, 46 | offval 7706 | . . 3 ⊢ (𝜑 → ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(1r‘𝑆)) = (𝑦 ∈ 𝐷 ↦ (𝑋(.r‘𝑅)if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )))) |
| 48 | ovif2 7532 | . . . . 5 ⊢ (𝑋(.r‘𝑅)if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )) = if(𝑦 = (𝐼 × {0}), (𝑋(.r‘𝑅)(1r‘𝑅)), (𝑋(.r‘𝑅) 0 )) | |
| 49 | 2, 18, 34, 5, 1 | ringridmd 20270 | . . . . . 6 ⊢ (𝜑 → (𝑋(.r‘𝑅)(1r‘𝑅)) = 𝑋) |
| 50 | 2, 18, 33, 5, 1 | ringrzd 20293 | . . . . . 6 ⊢ (𝜑 → (𝑋(.r‘𝑅) 0 ) = 0 ) |
| 51 | 49, 50 | ifeq12d 4547 | . . . . 5 ⊢ (𝜑 → if(𝑦 = (𝐼 × {0}), (𝑋(.r‘𝑅)(1r‘𝑅)), (𝑋(.r‘𝑅) 0 )) = if(𝑦 = (𝐼 × {0}), 𝑋, 0 )) |
| 52 | 48, 51 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → (𝑋(.r‘𝑅)if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )) = if(𝑦 = (𝐼 × {0}), 𝑋, 0 )) |
| 53 | 52 | mpteq2dv 5244 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ (𝑋(.r‘𝑅)if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
| 54 | 47, 53 | eqtrd 2777 | . 2 ⊢ (𝜑 → ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(1r‘𝑆)) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
| 55 | 16, 23, 54 | 3eqtrd 2781 | 1 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ifcif 4525 {csn 4626 ↦ cmpt 5225 × cxp 5683 ◡ccnv 5684 “ cima 5688 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 ↑m cmap 8866 Fincfn 8985 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 1rcur 20178 Ringcrg 20230 algSccascl 21872 mPwSer cmps 21924 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-mulg 19086 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-ascl 21875 df-psr 21929 |
| This theorem is referenced by: (None) |
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