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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 6614 | . . . . 5 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
2 | 1 | fveq2d 6549 | . . . 4 ⊢ (𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘𝑅)) |
3 | 2 | fveq2d 6549 | . . 3 ⊢ (𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅))) |
4 | eqid 2797 | . . . . . 6 ⊢ (Poly1‘∅) = (Poly1‘∅) | |
5 | eqid 2797 | . . . . . 6 ⊢ (1o mPoly ∅) = (1o mPoly ∅) | |
6 | eqid 2797 | . . . . . 6 ⊢ (+g‘(Poly1‘∅)) = (+g‘(Poly1‘∅)) | |
7 | 4, 5, 6 | ply1plusg 20080 | . . . . 5 ⊢ (+g‘(Poly1‘∅)) = (+g‘(1o mPoly ∅)) |
8 | eqid 2797 | . . . . . . 7 ⊢ (1o mPwSer ∅) = (1o mPwSer ∅) | |
9 | eqid 2797 | . . . . . . 7 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPoly ∅)) | |
10 | 5, 8, 9 | mplplusg 20075 | . . . . . 6 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPwSer ∅)) |
11 | base0 16369 | . . . . . . . . . 10 ⊢ ∅ = (Base‘∅) | |
12 | psr1baslem 20040 | . . . . . . . . . 10 ⊢ (ℕ0 ↑𝑚 1o) = {𝑎 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
13 | eqid 2797 | . . . . . . . . . 10 ⊢ (Base‘(1o mPwSer ∅)) = (Base‘(1o mPwSer ∅)) | |
14 | 1on 7967 | . . . . . . . . . . 11 ⊢ 1o ∈ On | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 1o ∈ On) |
16 | 8, 11, 12, 13, 15 | psrbas 19850 | . . . . . . . . 9 ⊢ (⊤ → (Base‘(1o mPwSer ∅)) = (∅ ↑𝑚 (ℕ0 ↑𝑚 1o))) |
17 | 16 | mptru 1532 | . . . . . . . 8 ⊢ (Base‘(1o mPwSer ∅)) = (∅ ↑𝑚 (ℕ0 ↑𝑚 1o)) |
18 | 0nn0 11766 | . . . . . . . . . . 11 ⊢ 0 ∈ ℕ0 | |
19 | 18 | fconst6 6444 | . . . . . . . . . 10 ⊢ (1o × {0}):1o⟶ℕ0 |
20 | nn0ex 11757 | . . . . . . . . . . 11 ⊢ ℕ0 ∈ V | |
21 | 1oex 7968 | . . . . . . . . . . 11 ⊢ 1o ∈ V | |
22 | 20, 21 | elmap 8292 | . . . . . . . . . 10 ⊢ ((1o × {0}) ∈ (ℕ0 ↑𝑚 1o) ↔ (1o × {0}):1o⟶ℕ0) |
23 | 19, 22 | mpbir 232 | . . . . . . . . 9 ⊢ (1o × {0}) ∈ (ℕ0 ↑𝑚 1o) |
24 | ne0i 4226 | . . . . . . . . 9 ⊢ ((1o × {0}) ∈ (ℕ0 ↑𝑚 1o) → (ℕ0 ↑𝑚 1o) ≠ ∅) | |
25 | map0b 8304 | . . . . . . . . 9 ⊢ ((ℕ0 ↑𝑚 1o) ≠ ∅ → (∅ ↑𝑚 (ℕ0 ↑𝑚 1o)) = ∅) | |
26 | 23, 24, 25 | mp2b 10 | . . . . . . . 8 ⊢ (∅ ↑𝑚 (ℕ0 ↑𝑚 1o)) = ∅ |
27 | 17, 26 | eqtr2i 2822 | . . . . . . 7 ⊢ ∅ = (Base‘(1o mPwSer ∅)) |
28 | eqid 2797 | . . . . . . 7 ⊢ (+g‘∅) = (+g‘∅) | |
29 | eqid 2797 | . . . . . . 7 ⊢ (+g‘(1o mPwSer ∅)) = (+g‘(1o mPwSer ∅)) | |
30 | 8, 27, 28, 29 | psrplusg 19853 | . . . . . 6 ⊢ (+g‘(1o mPwSer ∅)) = ( ∘𝑓 (+g‘∅) ↾ (∅ × ∅)) |
31 | xp0 5898 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
32 | 31 | reseq2i 5738 | . . . . . 6 ⊢ ( ∘𝑓 (+g‘∅) ↾ (∅ × ∅)) = ( ∘𝑓 (+g‘∅) ↾ ∅) |
33 | 10, 30, 32 | 3eqtri 2825 | . . . . 5 ⊢ (+g‘(1o mPoly ∅)) = ( ∘𝑓 (+g‘∅) ↾ ∅) |
34 | res0 5745 | . . . . . 6 ⊢ ( ∘𝑓 (+g‘∅) ↾ ∅) = ∅ | |
35 | df-plusg 16411 | . . . . . . 7 ⊢ +g = Slot 2 | |
36 | 35 | str0 16368 | . . . . . 6 ⊢ ∅ = (+g‘∅) |
37 | 34, 36 | eqtri 2821 | . . . . 5 ⊢ ( ∘𝑓 (+g‘∅) ↾ ∅) = (+g‘∅) |
38 | 7, 33, 37 | 3eqtri 2825 | . . . 4 ⊢ (+g‘(Poly1‘∅)) = (+g‘∅) |
39 | fvprc 6538 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → ( I ‘𝑅) = ∅) | |
40 | 39 | fveq2d 6549 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘∅)) |
41 | 40 | fveq2d 6549 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘∅))) |
42 | fvprc 6538 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
43 | 42 | fveq2d 6549 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘𝑅)) = (+g‘∅)) |
44 | 38, 41, 43 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅))) |
45 | 3, 44 | pm2.61i 183 | . 2 ⊢ (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅)) |
46 | 45 | eqcomi 2806 | 1 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘( I ‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1525 ⊤wtru 1526 ∈ wcel 2083 ≠ wne 2986 Vcvv 3440 ∅c0 4217 {csn 4478 I cid 5354 × cxp 5448 ↾ cres 5452 Oncon0 6073 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 ∘𝑓 cof 7272 1oc1o 7953 ↑𝑚 cmap 8263 0cc0 10390 2c2 11546 ℕ0cn0 11751 Basecbs 16316 +gcplusg 16398 mPwSer cmps 19823 mPoly cmpl 19825 Poly1cpl1 20032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-fz 12747 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-sca 16414 df-vsca 16415 df-tset 16417 df-ple 16418 df-psr 19828 df-mpl 19830 df-opsr 19832 df-psr1 20035 df-ply1 20037 |
This theorem is referenced by: (None) |
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