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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 6743 | . . . . 5 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
2 | 1 | fveq2d 6677 | . . . 4 ⊢ (𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘𝑅)) |
3 | 2 | fveq2d 6677 | . . 3 ⊢ (𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅))) |
4 | eqid 2824 | . . . . . 6 ⊢ (Poly1‘∅) = (Poly1‘∅) | |
5 | eqid 2824 | . . . . . 6 ⊢ (1o mPoly ∅) = (1o mPoly ∅) | |
6 | eqid 2824 | . . . . . 6 ⊢ (+g‘(Poly1‘∅)) = (+g‘(Poly1‘∅)) | |
7 | 4, 5, 6 | ply1plusg 20396 | . . . . 5 ⊢ (+g‘(Poly1‘∅)) = (+g‘(1o mPoly ∅)) |
8 | eqid 2824 | . . . . . . 7 ⊢ (1o mPwSer ∅) = (1o mPwSer ∅) | |
9 | eqid 2824 | . . . . . . 7 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPoly ∅)) | |
10 | 5, 8, 9 | mplplusg 20391 | . . . . . 6 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPwSer ∅)) |
11 | base0 16539 | . . . . . . . . . 10 ⊢ ∅ = (Base‘∅) | |
12 | psr1baslem 20356 | . . . . . . . . . 10 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
13 | eqid 2824 | . . . . . . . . . 10 ⊢ (Base‘(1o mPwSer ∅)) = (Base‘(1o mPwSer ∅)) | |
14 | 1on 8112 | . . . . . . . . . . 11 ⊢ 1o ∈ On | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 1o ∈ On) |
16 | 8, 11, 12, 13, 15 | psrbas 20161 | . . . . . . . . 9 ⊢ (⊤ → (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o))) |
17 | 16 | mptru 1543 | . . . . . . . 8 ⊢ (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o)) |
18 | 0nn0 11915 | . . . . . . . . . . 11 ⊢ 0 ∈ ℕ0 | |
19 | 18 | fconst6 6572 | . . . . . . . . . 10 ⊢ (1o × {0}):1o⟶ℕ0 |
20 | nn0ex 11906 | . . . . . . . . . . 11 ⊢ ℕ0 ∈ V | |
21 | 1oex 8113 | . . . . . . . . . . 11 ⊢ 1o ∈ V | |
22 | 20, 21 | elmap 8438 | . . . . . . . . . 10 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {0}):1o⟶ℕ0) |
23 | 19, 22 | mpbir 233 | . . . . . . . . 9 ⊢ (1o × {0}) ∈ (ℕ0 ↑m 1o) |
24 | ne0i 4303 | . . . . . . . . 9 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) → (ℕ0 ↑m 1o) ≠ ∅) | |
25 | map0b 8450 | . . . . . . . . 9 ⊢ ((ℕ0 ↑m 1o) ≠ ∅ → (∅ ↑m (ℕ0 ↑m 1o)) = ∅) | |
26 | 23, 24, 25 | mp2b 10 | . . . . . . . 8 ⊢ (∅ ↑m (ℕ0 ↑m 1o)) = ∅ |
27 | 17, 26 | eqtr2i 2848 | . . . . . . 7 ⊢ ∅ = (Base‘(1o mPwSer ∅)) |
28 | eqid 2824 | . . . . . . 7 ⊢ (+g‘∅) = (+g‘∅) | |
29 | eqid 2824 | . . . . . . 7 ⊢ (+g‘(1o mPwSer ∅)) = (+g‘(1o mPwSer ∅)) | |
30 | 8, 27, 28, 29 | psrplusg 20164 | . . . . . 6 ⊢ (+g‘(1o mPwSer ∅)) = ( ∘f (+g‘∅) ↾ (∅ × ∅)) |
31 | xp0 6018 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
32 | 31 | reseq2i 5853 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ (∅ × ∅)) = ( ∘f (+g‘∅) ↾ ∅) |
33 | 10, 30, 32 | 3eqtri 2851 | . . . . 5 ⊢ (+g‘(1o mPoly ∅)) = ( ∘f (+g‘∅) ↾ ∅) |
34 | res0 5860 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ ∅) = ∅ | |
35 | df-plusg 16581 | . . . . . . 7 ⊢ +g = Slot 2 | |
36 | 35 | str0 16538 | . . . . . 6 ⊢ ∅ = (+g‘∅) |
37 | 34, 36 | eqtri 2847 | . . . . 5 ⊢ ( ∘f (+g‘∅) ↾ ∅) = (+g‘∅) |
38 | 7, 33, 37 | 3eqtri 2851 | . . . 4 ⊢ (+g‘(Poly1‘∅)) = (+g‘∅) |
39 | fvprc 6666 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → ( I ‘𝑅) = ∅) | |
40 | 39 | fveq2d 6677 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘∅)) |
41 | 40 | fveq2d 6677 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘∅))) |
42 | fvprc 6666 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
43 | 42 | fveq2d 6677 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘𝑅)) = (+g‘∅)) |
44 | 38, 41, 43 | 3eqtr4a 2885 | . . 3 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅))) |
45 | 3, 44 | pm2.61i 184 | . 2 ⊢ (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅)) |
46 | 45 | eqcomi 2833 | 1 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘( I ‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ⊤wtru 1537 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∅c0 4294 {csn 4570 I cid 5462 × cxp 5556 ↾ cres 5560 Oncon0 6194 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ∘f cof 7410 1oc1o 8098 ↑m cmap 8409 0cc0 10540 2c2 11695 ℕ0cn0 11900 Basecbs 16486 +gcplusg 16568 mPwSer cmps 20134 mPoly cmpl 20136 Poly1cpl1 20348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 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-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-tset 16587 df-ple 16588 df-psr 20139 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-ply1 20353 |
This theorem is referenced by: (None) |
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