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Mirrors > Home > MPE Home > Th. List > Mathboxes > selvcllem2 | Structured version Visualization version GIF version |
Description: 𝐷 is a ring homomorphism. (Contributed by SN, 15-Dec-2023.) |
Ref | Expression |
---|---|
selvcllem2.u | ⊢ 𝑈 = (𝐼 mPoly 𝑅) |
selvcllem2.t | ⊢ 𝑇 = (𝐽 mPoly 𝑈) |
selvcllem2.c | ⊢ 𝐶 = (algSc‘𝑇) |
selvcllem2.d | ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) |
selvcllem2.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
selvcllem2.j | ⊢ (𝜑 → 𝐽 ∈ 𝑊) |
selvcllem2.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
Ref | Expression |
---|---|
selvcllem2 | ⊢ (𝜑 → 𝐷 ∈ (𝑅 RingHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | selvcllem2.d | . 2 ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) | |
2 | selvcllem2.u | . . . . . . 7 ⊢ 𝑈 = (𝐼 mPoly 𝑅) | |
3 | selvcllem2.t | . . . . . . 7 ⊢ 𝑇 = (𝐽 mPoly 𝑈) | |
4 | selvcllem2.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
5 | selvcllem2.j | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝑊) | |
6 | selvcllem2.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
7 | 2, 3, 4, 5, 6 | selvcllem1 41452 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ AssAlg) |
8 | selvcllem2.c | . . . . . . 7 ⊢ 𝐶 = (algSc‘𝑇) | |
9 | eqid 2731 | . . . . . . 7 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
10 | 8, 9 | asclrhm 21664 | . . . . . 6 ⊢ (𝑇 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑇) RingHom 𝑇)) |
11 | 7, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑇) RingHom 𝑇)) |
12 | 2 | mplassa 21801 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing) → 𝑈 ∈ AssAlg) |
13 | 4, 6, 12 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ AssAlg) |
14 | 3, 5, 13 | mplsca 21792 | . . . . . 6 ⊢ (𝜑 → 𝑈 = (Scalar‘𝑇)) |
15 | 14 | oveq1d 7427 | . . . . 5 ⊢ (𝜑 → (𝑈 RingHom 𝑇) = ((Scalar‘𝑇) RingHom 𝑇)) |
16 | 11, 15 | eleqtrrd 2835 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝑈 RingHom 𝑇)) |
17 | eqid 2731 | . . . . . 6 ⊢ (algSc‘𝑈) = (algSc‘𝑈) | |
18 | eqid 2731 | . . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
19 | 17, 18 | asclrhm 21664 | . . . . 5 ⊢ (𝑈 ∈ AssAlg → (algSc‘𝑈) ∈ ((Scalar‘𝑈) RingHom 𝑈)) |
20 | 13, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (algSc‘𝑈) ∈ ((Scalar‘𝑈) RingHom 𝑈)) |
21 | rhmco 20393 | . . . 4 ⊢ ((𝐶 ∈ (𝑈 RingHom 𝑇) ∧ (algSc‘𝑈) ∈ ((Scalar‘𝑈) RingHom 𝑈)) → (𝐶 ∘ (algSc‘𝑈)) ∈ ((Scalar‘𝑈) RingHom 𝑇)) | |
22 | 16, 20, 21 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝐶 ∘ (algSc‘𝑈)) ∈ ((Scalar‘𝑈) RingHom 𝑇)) |
23 | 2, 4, 6 | mplsca 21792 | . . . 4 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈)) |
24 | 23 | oveq1d 7427 | . . 3 ⊢ (𝜑 → (𝑅 RingHom 𝑇) = ((Scalar‘𝑈) RingHom 𝑇)) |
25 | 22, 24 | eleqtrrd 2835 | . 2 ⊢ (𝜑 → (𝐶 ∘ (algSc‘𝑈)) ∈ (𝑅 RingHom 𝑇)) |
26 | 1, 25 | eqeltrid 2836 | 1 ⊢ (𝜑 → 𝐷 ∈ (𝑅 RingHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7412 Scalarcsca 17205 CRingccrg 20129 RingHom crh 20361 AssAlgcasa 21625 algSccascl 21627 mPoly cmpl 21679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-fzo 13633 df-seq 13972 df-hash 14296 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-lmod 20617 df-lss 20688 df-assa 21628 df-ascl 21630 df-psr 21682 df-mpl 21684 |
This theorem is referenced by: selvcllem3 41454 selvcllem4 41456 selvadd 41463 selvmul 41464 |
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