![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subrgascl | Structured version Visualization version GIF version |
Description: The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
subrgascl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
subrgascl.a | ⊢ 𝐴 = (algSc‘𝑃) |
subrgascl.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
subrgascl.u | ⊢ 𝑈 = (𝐼 mPoly 𝐻) |
subrgascl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
subrgascl.r | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
subrgascl.c | ⊢ 𝐶 = (algSc‘𝑈) |
Ref | Expression |
---|---|
subrgascl | ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgascl.c | . . . 4 ⊢ 𝐶 = (algSc‘𝑈) | |
2 | eqid 2735 | . . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
3 | eqid 2735 | . . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
4 | 1, 2, 3 | asclfn 21919 | . . 3 ⊢ 𝐶 Fn (Base‘(Scalar‘𝑈)) |
5 | subrgascl.r | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
6 | subrgascl.h | . . . . . . 7 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
7 | 6 | subrgbas 20598 | . . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 = (Base‘𝐻)) |
8 | 5, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑇 = (Base‘𝐻)) |
9 | subrgascl.u | . . . . . . 7 ⊢ 𝑈 = (𝐼 mPoly 𝐻) | |
10 | subrgascl.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
11 | 6 | ovexi 7465 | . . . . . . . 8 ⊢ 𝐻 ∈ V |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ V) |
13 | 9, 10, 12 | mplsca 22051 | . . . . . 6 ⊢ (𝜑 → 𝐻 = (Scalar‘𝑈)) |
14 | 13 | fveq2d 6911 | . . . . 5 ⊢ (𝜑 → (Base‘𝐻) = (Base‘(Scalar‘𝑈))) |
15 | 8, 14 | eqtrd 2775 | . . . 4 ⊢ (𝜑 → 𝑇 = (Base‘(Scalar‘𝑈))) |
16 | 15 | fneq2d 6663 | . . 3 ⊢ (𝜑 → (𝐶 Fn 𝑇 ↔ 𝐶 Fn (Base‘(Scalar‘𝑈)))) |
17 | 4, 16 | mpbiri 258 | . 2 ⊢ (𝜑 → 𝐶 Fn 𝑇) |
18 | subrgascl.a | . . . . 5 ⊢ 𝐴 = (algSc‘𝑃) | |
19 | eqid 2735 | . . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
20 | eqid 2735 | . . . . 5 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
21 | 18, 19, 20 | asclfn 21919 | . . . 4 ⊢ 𝐴 Fn (Base‘(Scalar‘𝑃)) |
22 | subrgascl.p | . . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
23 | subrgrcl 20593 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring) | |
24 | 5, 23 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
25 | 22, 10, 24 | mplsca 22051 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
26 | 25 | fveq2d 6911 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
27 | 26 | fneq2d 6663 | . . . 4 ⊢ (𝜑 → (𝐴 Fn (Base‘𝑅) ↔ 𝐴 Fn (Base‘(Scalar‘𝑃)))) |
28 | 21, 27 | mpbiri 258 | . . 3 ⊢ (𝜑 → 𝐴 Fn (Base‘𝑅)) |
29 | eqid 2735 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
30 | 29 | subrgss 20589 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ⊆ (Base‘𝑅)) |
31 | 5, 30 | syl 17 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝑅)) |
32 | fnssres 6692 | . . 3 ⊢ ((𝐴 Fn (Base‘𝑅) ∧ 𝑇 ⊆ (Base‘𝑅)) → (𝐴 ↾ 𝑇) Fn 𝑇) | |
33 | 28, 31, 32 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 ↾ 𝑇) Fn 𝑇) |
34 | fvres 6926 | . . . 4 ⊢ (𝑥 ∈ 𝑇 → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥)) | |
35 | 34 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → ((𝐴 ↾ 𝑇)‘𝑥) = (𝐴‘𝑥)) |
36 | eqid 2735 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
37 | 6, 36 | subrg0 20596 | . . . . . . . 8 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0g‘𝑅) = (0g‘𝐻)) |
38 | 5, 37 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) = (0g‘𝐻)) |
39 | 38 | ifeq2d 4551 | . . . . . 6 ⊢ (𝜑 → if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻))) |
40 | 39 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)) = if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻))) |
41 | 40 | mpteq2dv 5250 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅))) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻)))) |
42 | eqid 2735 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
43 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐼 ∈ 𝑊) |
44 | 24 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑅 ∈ Ring) |
45 | 31 | sselda 3995 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝑅)) |
46 | 22, 42, 36, 29, 18, 43, 44, 45 | mplascl 22106 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝑅)))) |
47 | eqid 2735 | . . . . 5 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
48 | eqid 2735 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
49 | 6 | subrgring 20591 | . . . . . . 7 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring) |
50 | 5, 49 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Ring) |
51 | 50 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝐻 ∈ Ring) |
52 | 8 | eleq2d 2825 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑇 ↔ 𝑥 ∈ (Base‘𝐻))) |
53 | 52 | biimpa 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (Base‘𝐻)) |
54 | 9, 42, 47, 48, 1, 43, 51, 53 | mplascl 22106 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = (𝐼 × {0}), 𝑥, (0g‘𝐻)))) |
55 | 41, 46, 54 | 3eqtr4d 2785 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐴‘𝑥) = (𝐶‘𝑥)) |
56 | 35, 55 | eqtr2d 2776 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝐶‘𝑥) = ((𝐴 ↾ 𝑇)‘𝑥)) |
57 | 17, 33, 56 | eqfnfvd 7054 | 1 ⊢ (𝜑 → 𝐶 = (𝐴 ↾ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ⊆ wss 3963 ifcif 4531 {csn 4631 ↦ cmpt 5231 × cxp 5687 ◡ccnv 5688 ↾ cres 5691 “ cima 5692 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 0cc0 11153 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 ↾s cress 17274 Scalarcsca 17301 0gc0g 17486 Ringcrg 20251 SubRingcsubrg 20586 algSccascl 21890 mPoly cmpl 21944 |
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-subg 19154 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-subrng 20563 df-subrg 20587 df-ascl 21893 df-psr 21947 df-mpl 21949 |
This theorem is referenced by: subrgasclcl 22109 subrg1ascl 22278 |
Copyright terms: Public domain | W3C validator |