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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 21985 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑆)) |
7 | 6 | fveq2d 6911 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑆))) |
8 | 2, 7 | eqtrid 2787 | . . . 4 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑆))) |
9 | 1, 8 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑆))) |
10 | psrascl.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑆) | |
11 | eqid 2735 | . . . 4 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
12 | eqid 2735 | . . . 4 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆)) | |
13 | eqid 2735 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
14 | eqid 2735 | . . . 4 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
15 | 10, 11, 12, 13, 14 | asclval 21918 | . . 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 2735 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
18 | eqid 2735 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
19 | psrascl.d | . . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
20 | 3, 4, 5 | psrring 22008 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring) |
21 | 17, 14 | ringidcl 20280 | . . . 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 21987 | . 2 ⊢ (𝜑 → (𝑋( ·𝑠 ‘𝑆)(1r‘𝑆)) = ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(1r‘𝑆))) |
24 | fnconstg 6797 | . . . . 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 21972 | . . . . 5 ⊢ (𝜑 → (1r‘𝑆):𝐷⟶𝐾) |
27 | 26 | ffnd 6738 | . . . 4 ⊢ (𝜑 → (1r‘𝑆) Fn 𝐷) |
28 | ovexd 7466 | . . . . 5 ⊢ (𝜑 → (ℕ0 ↑m 𝐼) ∈ V) | |
29 | 19, 28 | rabexd 5346 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
30 | inidm 4235 | . . . 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 2735 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
35 | 3, 4, 5, 19, 33, 34, 14 | psr1 22009 | . . . . . . 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 6909 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ((1r‘𝑆)‘𝑦) = ((𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 ))‘𝑦)) |
38 | eqeq1 2739 | . . . . . . . 8 ⊢ (𝑑 = 𝑦 → (𝑑 = (𝐼 × {0}) ↔ 𝑦 = (𝐼 × {0}))) | |
39 | 38 | ifbid 4554 | . . . . . . 7 ⊢ (𝑑 = 𝑦 → if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 ) = if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )) |
40 | eqid 2735 | . . . . . . 7 ⊢ (𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 )) = (𝑑 ∈ 𝐷 ↦ if(𝑑 = (𝐼 × {0}), (1r‘𝑅), 0 )) | |
41 | fvex 6920 | . . . . . . . 8 ⊢ (1r‘𝑅) ∈ V | |
42 | 33 | fvexi 6921 | . . . . . . . 8 ⊢ 0 ∈ V |
43 | 41, 42 | ifex 4581 | . . . . . . 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 2775 | . . . 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 20287 | . . . . . 6 ⊢ (𝜑 → (𝑋(.r‘𝑅)(1r‘𝑅)) = 𝑋) |
50 | 2, 18, 33, 5, 1 | ringrzd 20310 | . . . . . 6 ⊢ (𝜑 → (𝑋(.r‘𝑅) 0 ) = 0 ) |
51 | 49, 50 | ifeq12d 4552 | . . . . 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 2787 | . . . 4 ⊢ (𝜑 → (𝑋(.r‘𝑅)if(𝑦 = (𝐼 × {0}), (1r‘𝑅), 0 )) = if(𝑦 = (𝐼 × {0}), 𝑋, 0 )) |
53 | 52 | mpteq2dv 5250 | . . 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 2775 | . 2 ⊢ (𝜑 → ((𝐷 × {𝑋}) ∘f (.r‘𝑅)(1r‘𝑆)) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 ))) |
55 | 16, 23, 54 | 3eqtrd 2779 | 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 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ifcif 4531 {csn 4631 ↦ cmpt 5231 × cxp 5687 ◡ccnv 5688 “ cima 5692 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 ↑m cmap 8865 Fincfn 8984 0cc0 11153 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 1rcur 20199 Ringcrg 20251 algSccascl 21890 mPwSer cmps 21942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-mulg 19099 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-ascl 21893 df-psr 21947 |
This theorem is referenced by: (None) |
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