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Mirrors > Home > MPE Home > Th. List > ply1plusgfvi | Structured version Visualization version GIF version |
Description: Protection compatibility of the univariate polynomial addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) |
Ref | Expression |
---|---|
ply1plusgfvi | ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘( I ‘𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvi 6980 | . . . . 5 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
2 | 1 | fveq2d 6907 | . . . 4 ⊢ (𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘𝑅)) |
3 | 2 | fveq2d 6907 | . . 3 ⊢ (𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅))) |
4 | eqid 2726 | . . . . . 6 ⊢ (Poly1‘∅) = (Poly1‘∅) | |
5 | eqid 2726 | . . . . . 6 ⊢ (1o mPoly ∅) = (1o mPoly ∅) | |
6 | eqid 2726 | . . . . . 6 ⊢ (+g‘(Poly1‘∅)) = (+g‘(Poly1‘∅)) | |
7 | 4, 5, 6 | ply1plusg 22215 | . . . . 5 ⊢ (+g‘(Poly1‘∅)) = (+g‘(1o mPoly ∅)) |
8 | eqid 2726 | . . . . . . 7 ⊢ (1o mPwSer ∅) = (1o mPwSer ∅) | |
9 | eqid 2726 | . . . . . . 7 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPoly ∅)) | |
10 | 5, 8, 9 | mplplusg 22018 | . . . . . 6 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPwSer ∅)) |
11 | base0 17220 | . . . . . . . . . 10 ⊢ ∅ = (Base‘∅) | |
12 | psr1baslem 22176 | . . . . . . . . . 10 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
13 | eqid 2726 | . . . . . . . . . 10 ⊢ (Base‘(1o mPwSer ∅)) = (Base‘(1o mPwSer ∅)) | |
14 | 1on 8510 | . . . . . . . . . . 11 ⊢ 1o ∈ On | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 1o ∈ On) |
16 | 8, 11, 12, 13, 15 | psrbas 21944 | . . . . . . . . 9 ⊢ (⊤ → (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o))) |
17 | 16 | mptru 1541 | . . . . . . . 8 ⊢ (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o)) |
18 | 0nn0 12541 | . . . . . . . . . . 11 ⊢ 0 ∈ ℕ0 | |
19 | 18 | fconst6 6794 | . . . . . . . . . 10 ⊢ (1o × {0}):1o⟶ℕ0 |
20 | nn0ex 12532 | . . . . . . . . . . 11 ⊢ ℕ0 ∈ V | |
21 | 1oex 8508 | . . . . . . . . . . 11 ⊢ 1o ∈ V | |
22 | 20, 21 | elmap 8902 | . . . . . . . . . 10 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {0}):1o⟶ℕ0) |
23 | 19, 22 | mpbir 230 | . . . . . . . . 9 ⊢ (1o × {0}) ∈ (ℕ0 ↑m 1o) |
24 | ne0i 4337 | . . . . . . . . 9 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) → (ℕ0 ↑m 1o) ≠ ∅) | |
25 | map0b 8914 | . . . . . . . . 9 ⊢ ((ℕ0 ↑m 1o) ≠ ∅ → (∅ ↑m (ℕ0 ↑m 1o)) = ∅) | |
26 | 23, 24, 25 | mp2b 10 | . . . . . . . 8 ⊢ (∅ ↑m (ℕ0 ↑m 1o)) = ∅ |
27 | 17, 26 | eqtr2i 2755 | . . . . . . 7 ⊢ ∅ = (Base‘(1o mPwSer ∅)) |
28 | eqid 2726 | . . . . . . 7 ⊢ (+g‘∅) = (+g‘∅) | |
29 | eqid 2726 | . . . . . . 7 ⊢ (+g‘(1o mPwSer ∅)) = (+g‘(1o mPwSer ∅)) | |
30 | 8, 27, 28, 29 | psrplusg 21947 | . . . . . 6 ⊢ (+g‘(1o mPwSer ∅)) = ( ∘f (+g‘∅) ↾ (∅ × ∅)) |
31 | xp0 6171 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
32 | 31 | reseq2i 5988 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ (∅ × ∅)) = ( ∘f (+g‘∅) ↾ ∅) |
33 | 10, 30, 32 | 3eqtri 2758 | . . . . 5 ⊢ (+g‘(1o mPoly ∅)) = ( ∘f (+g‘∅) ↾ ∅) |
34 | res0 5995 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ ∅) = ∅ | |
35 | plusgid 17295 | . . . . . . 7 ⊢ +g = Slot (+g‘ndx) | |
36 | 35 | str0 17193 | . . . . . 6 ⊢ ∅ = (+g‘∅) |
37 | 34, 36 | eqtri 2754 | . . . . 5 ⊢ ( ∘f (+g‘∅) ↾ ∅) = (+g‘∅) |
38 | 7, 33, 37 | 3eqtri 2758 | . . . 4 ⊢ (+g‘(Poly1‘∅)) = (+g‘∅) |
39 | fvprc 6895 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → ( I ‘𝑅) = ∅) | |
40 | 39 | fveq2d 6907 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘∅)) |
41 | 40 | fveq2d 6907 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘∅))) |
42 | fvprc 6895 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
43 | 42 | fveq2d 6907 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘𝑅)) = (+g‘∅)) |
44 | 38, 41, 43 | 3eqtr4a 2792 | . . 3 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅))) |
45 | 3, 44 | pm2.61i 182 | . 2 ⊢ (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅)) |
46 | 45 | eqcomi 2735 | 1 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘( I ‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ∅c0 4325 {csn 4633 I cid 5581 × cxp 5682 ↾ cres 5686 Oncon0 6378 ⟶wf 6552 ‘cfv 6556 (class class class)co 7426 ∘f cof 7690 1oc1o 8491 ↑m cmap 8857 0cc0 11160 ℕ0cn0 12526 ndxcnx 17197 Basecbs 17215 +gcplusg 17268 mPwSer cmps 21903 mPoly cmpl 21905 Poly1cpl1 22168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-supp 8177 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9408 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-fz 13541 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-tset 17287 df-ple 17288 df-psr 21908 df-mpl 21910 df-opsr 21912 df-psr1 22171 df-ply1 22173 |
This theorem is referenced by: (None) |
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