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Mirrors > Home > MPE Home > Th. List > ressply1mul | Structured version Visualization version GIF version |
Description: A restricted polynomial algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.) |
Ref | Expression |
---|---|
ressply1.s | ⊢ 𝑆 = (Poly1‘𝑅) |
ressply1.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
ressply1.u | ⊢ 𝑈 = (Poly1‘𝐻) |
ressply1.b | ⊢ 𝐵 = (Base‘𝑈) |
ressply1.2 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
ressply1.p | ⊢ 𝑃 = (𝑆 ↾s 𝐵) |
Ref | Expression |
---|---|
ressply1mul | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | ressply1.h | . . 3 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
3 | eqid 2823 | . . 3 ⊢ (1o mPoly 𝐻) = (1o mPoly 𝐻) | |
4 | ressply1.u | . . . 4 ⊢ 𝑈 = (Poly1‘𝐻) | |
5 | eqid 2823 | . . . 4 ⊢ (PwSer1‘𝐻) = (PwSer1‘𝐻) | |
6 | ressply1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑈) | |
7 | 4, 5, 6 | ply1bas 20365 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝐻)) |
8 | 1on 8111 | . . . 4 ⊢ 1o ∈ On | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ On) |
10 | ressply1.2 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
11 | eqid 2823 | . . 3 ⊢ ((1o mPoly 𝑅) ↾s 𝐵) = ((1o mPoly 𝑅) ↾s 𝐵) | |
12 | 1, 2, 3, 7, 9, 10, 11 | ressmplmul 20241 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘(1o mPoly 𝐻))𝑌) = (𝑋(.r‘((1o mPoly 𝑅) ↾s 𝐵))𝑌)) |
13 | eqid 2823 | . . . 4 ⊢ (.r‘𝑈) = (.r‘𝑈) | |
14 | 4, 3, 13 | ply1mulr 20397 | . . 3 ⊢ (.r‘𝑈) = (.r‘(1o mPoly 𝐻)) |
15 | 14 | oveqi 7171 | . 2 ⊢ (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘(1o mPoly 𝐻))𝑌) |
16 | ressply1.s | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
17 | eqid 2823 | . . . . 5 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
18 | 16, 1, 17 | ply1mulr 20397 | . . . 4 ⊢ (.r‘𝑆) = (.r‘(1o mPoly 𝑅)) |
19 | 6 | fvexi 6686 | . . . . 5 ⊢ 𝐵 ∈ V |
20 | ressply1.p | . . . . . 6 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
21 | 20, 17 | ressmulr 16627 | . . . . 5 ⊢ (𝐵 ∈ V → (.r‘𝑆) = (.r‘𝑃)) |
22 | 19, 21 | ax-mp 5 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑃) |
23 | eqid 2823 | . . . . . 6 ⊢ (.r‘(1o mPoly 𝑅)) = (.r‘(1o mPoly 𝑅)) | |
24 | 11, 23 | ressmulr 16627 | . . . . 5 ⊢ (𝐵 ∈ V → (.r‘(1o mPoly 𝑅)) = (.r‘((1o mPoly 𝑅) ↾s 𝐵))) |
25 | 19, 24 | ax-mp 5 | . . . 4 ⊢ (.r‘(1o mPoly 𝑅)) = (.r‘((1o mPoly 𝑅) ↾s 𝐵)) |
26 | 18, 22, 25 | 3eqtr3i 2854 | . . 3 ⊢ (.r‘𝑃) = (.r‘((1o mPoly 𝑅) ↾s 𝐵)) |
27 | 26 | oveqi 7171 | . 2 ⊢ (𝑋(.r‘𝑃)𝑌) = (𝑋(.r‘((1o mPoly 𝑅) ↾s 𝐵))𝑌) |
28 | 12, 15, 27 | 3eqtr4g 2883 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(.r‘𝑈)𝑌) = (𝑋(.r‘𝑃)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 Oncon0 6193 ‘cfv 6357 (class class class)co 7158 1oc1o 8097 Basecbs 16485 ↾s cress 16486 .rcmulr 16568 SubRingcsubrg 19533 mPoly cmpl 20135 PwSer1cps1 20345 Poly1cpl1 20347 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-seq 13373 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-subg 18278 df-mgp 19242 df-ring 19301 df-subrg 19535 df-psr 20138 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-ply1 20352 |
This theorem is referenced by: (None) |
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