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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 2610 | . . 3 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
2 | ressply1.h | . . 3 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
3 | eqid 2610 | . . 3 ⊢ (1𝑜 mPoly 𝐻) = (1𝑜 mPoly 𝐻) | |
4 | ressply1.u | . . . 4 ⊢ 𝑈 = (Poly1‘𝐻) | |
5 | eqid 2610 | . . . 4 ⊢ (PwSer1‘𝐻) = (PwSer1‘𝐻) | |
6 | ressply1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑈) | |
7 | 4, 5, 6 | ply1bas 19386 | . . 3 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝐻)) |
8 | 1on 7454 | . . . 4 ⊢ 1𝑜 ∈ On | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 1𝑜 ∈ On) |
10 | ressply1.2 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
11 | eqid 2610 | . . 3 ⊢ ((1𝑜 mPoly 𝑅) ↾s 𝐵) = ((1𝑜 mPoly 𝑅) ↾s 𝐵) | |
12 | 1, 2, 3, 7, 9, 10, 11 | ressmpladd 19278 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(+g‘(1𝑜 mPoly 𝐻))𝑌) = (𝑋(+g‘((1𝑜 mPoly 𝑅) ↾s 𝐵))𝑌)) |
13 | eqid 2610 | . . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
14 | 4, 3, 13 | ply1plusg 19416 | . . 3 ⊢ (+g‘𝑈) = (+g‘(1𝑜 mPoly 𝐻)) |
15 | 14 | oveqi 6562 | . 2 ⊢ (𝑋(+g‘𝑈)𝑌) = (𝑋(+g‘(1𝑜 mPoly 𝐻))𝑌) |
16 | ressply1.s | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
17 | eqid 2610 | . . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆) | |
18 | 16, 1, 17 | ply1plusg 19416 | . . . 4 ⊢ (+g‘𝑆) = (+g‘(1𝑜 mPoly 𝑅)) |
19 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝑈) ∈ V | |
20 | 6, 19 | eqeltri 2684 | . . . . 5 ⊢ 𝐵 ∈ V |
21 | ressply1.p | . . . . . 6 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
22 | 21, 17 | ressplusg 15818 | . . . . 5 ⊢ (𝐵 ∈ V → (+g‘𝑆) = (+g‘𝑃)) |
23 | 20, 22 | ax-mp 5 | . . . 4 ⊢ (+g‘𝑆) = (+g‘𝑃) |
24 | eqid 2610 | . . . . . 6 ⊢ (+g‘(1𝑜 mPoly 𝑅)) = (+g‘(1𝑜 mPoly 𝑅)) | |
25 | 11, 24 | ressplusg 15818 | . . . . 5 ⊢ (𝐵 ∈ V → (+g‘(1𝑜 mPoly 𝑅)) = (+g‘((1𝑜 mPoly 𝑅) ↾s 𝐵))) |
26 | 20, 25 | ax-mp 5 | . . . 4 ⊢ (+g‘(1𝑜 mPoly 𝑅)) = (+g‘((1𝑜 mPoly 𝑅) ↾s 𝐵)) |
27 | 18, 23, 26 | 3eqtr3i 2640 | . . 3 ⊢ (+g‘𝑃) = (+g‘((1𝑜 mPoly 𝑅) ↾s 𝐵)) |
28 | 27 | oveqi 6562 | . 2 ⊢ (𝑋(+g‘𝑃)𝑌) = (𝑋(+g‘((1𝑜 mPoly 𝑅) ↾s 𝐵))𝑌) |
29 | 12, 15, 28 | 3eqtr4g 2669 | 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 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 Oncon0 5640 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 Basecbs 15695 ↾s cress 15696 +gcplusg 15768 SubRingcsubrg 18599 mPoly cmpl 19174 PwSer1cps1 19366 Poly1cpl1 19368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-subg 17414 df-ring 18372 df-subrg 18601 df-psr 19177 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-ply1 19373 |
This theorem is referenced by: gsumply1subr 19425 |
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