Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ressply1add | Structured version Visualization version GIF version |
Description: A restricted polynomial algebra has the same addition 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 |
---|---|
ressply1add | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘𝑈)𝑌) = (𝑋(+g‘𝑃)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | ressply1.h | . . 3 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
3 | eqid 2818 | . . 3 ⊢ (1o mPoly 𝐻) = (1o mPoly 𝐻) | |
4 | ressply1.u | . . . 4 ⊢ 𝑈 = (Poly1‘𝐻) | |
5 | eqid 2818 | . . . 4 ⊢ (PwSer1‘𝐻) = (PwSer1‘𝐻) | |
6 | ressply1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑈) | |
7 | 4, 5, 6 | ply1bas 20291 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝐻)) |
8 | 1on 8098 | . . . 4 ⊢ 1o ∈ On | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 1o ∈ On) |
10 | ressply1.2 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
11 | eqid 2818 | . . 3 ⊢ ((1o mPoly 𝑅) ↾s 𝐵) = ((1o mPoly 𝑅) ↾s 𝐵) | |
12 | 1, 2, 3, 7, 9, 10, 11 | ressmpladd 20166 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘(1o mPoly 𝐻))𝑌) = (𝑋(+g‘((1o mPoly 𝑅) ↾s 𝐵))𝑌)) |
13 | eqid 2818 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
14 | 4, 3, 13 | ply1plusg 20321 | . . 3 ⊢ (+g‘𝑈) = (+g‘(1o mPoly 𝐻)) |
15 | 14 | oveqi 7158 | . 2 ⊢ (𝑋(+g‘𝑈)𝑌) = (𝑋(+g‘(1o mPoly 𝐻))𝑌) |
16 | ressply1.s | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
17 | eqid 2818 | . . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
18 | 16, 1, 17 | ply1plusg 20321 | . . . 4 ⊢ (+g‘𝑆) = (+g‘(1o mPoly 𝑅)) |
19 | 6 | fvexi 6677 | . . . . 5 ⊢ 𝐵 ∈ V |
20 | ressply1.p | . . . . . 6 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
21 | 20, 17 | ressplusg 16600 | . . . . 5 ⊢ (𝐵 ∈ V → (+g‘𝑆) = (+g‘𝑃)) |
22 | 19, 21 | ax-mp 5 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑃) |
23 | eqid 2818 | . . . . . 6 ⊢ (+g‘(1o mPoly 𝑅)) = (+g‘(1o mPoly 𝑅)) | |
24 | 11, 23 | ressplusg 16600 | . . . . 5 ⊢ (𝐵 ∈ V → (+g‘(1o mPoly 𝑅)) = (+g‘((1o mPoly 𝑅) ↾s 𝐵))) |
25 | 19, 24 | ax-mp 5 | . . . 4 ⊢ (+g‘(1o mPoly 𝑅)) = (+g‘((1o mPoly 𝑅) ↾s 𝐵)) |
26 | 18, 22, 25 | 3eqtr3i 2849 | . . 3 ⊢ (+g‘𝑃) = (+g‘((1o mPoly 𝑅) ↾s 𝐵)) |
27 | 26 | oveqi 7158 | . 2 ⊢ (𝑋(+g‘𝑃)𝑌) = (𝑋(+g‘((1o mPoly 𝑅) ↾s 𝐵))𝑌) |
28 | 12, 15, 27 | 3eqtr4g 2878 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘𝑈)𝑌) = (𝑋(+g‘𝑃)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 Oncon0 6184 ‘cfv 6348 (class class class)co 7145 1oc1o 8084 Basecbs 16471 ↾s cress 16472 +gcplusg 16553 SubRingcsubrg 19460 mPoly cmpl 20061 PwSer1cps1 20271 Poly1cpl1 20273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-subg 18214 df-ring 19228 df-subrg 19462 df-psr 20064 df-mpl 20066 df-opsr 20068 df-psr1 20276 df-ply1 20278 |
This theorem is referenced by: gsumply1subr 20330 |
Copyright terms: Public domain | W3C validator |