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| 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 6984 | . . . . 5 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
| 2 | 1 | fveq2d 6909 | . . . 4 ⊢ (𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘𝑅)) | 
| 3 | 2 | fveq2d 6909 | . . 3 ⊢ (𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅))) | 
| 4 | eqid 2736 | . . . . . 6 ⊢ (Poly1‘∅) = (Poly1‘∅) | |
| 5 | eqid 2736 | . . . . . 6 ⊢ (1o mPoly ∅) = (1o mPoly ∅) | |
| 6 | eqid 2736 | . . . . . 6 ⊢ (+g‘(Poly1‘∅)) = (+g‘(Poly1‘∅)) | |
| 7 | 4, 5, 6 | ply1plusg 22226 | . . . . 5 ⊢ (+g‘(Poly1‘∅)) = (+g‘(1o mPoly ∅)) | 
| 8 | eqid 2736 | . . . . . . 7 ⊢ (1o mPwSer ∅) = (1o mPwSer ∅) | |
| 9 | eqid 2736 | . . . . . . 7 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPoly ∅)) | |
| 10 | 5, 8, 9 | mplplusg 22028 | . . . . . 6 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPwSer ∅)) | 
| 11 | base0 17253 | . . . . . . . . . 10 ⊢ ∅ = (Base‘∅) | |
| 12 | psr1baslem 22187 | . . . . . . . . . 10 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
| 13 | eqid 2736 | . . . . . . . . . 10 ⊢ (Base‘(1o mPwSer ∅)) = (Base‘(1o mPwSer ∅)) | |
| 14 | 1on 8519 | . . . . . . . . . . 11 ⊢ 1o ∈ On | |
| 15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 1o ∈ On) | 
| 16 | 8, 11, 12, 13, 15 | psrbas 21954 | . . . . . . . . 9 ⊢ (⊤ → (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o))) | 
| 17 | 16 | mptru 1546 | . . . . . . . 8 ⊢ (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o)) | 
| 18 | 0nn0 12543 | . . . . . . . . . . 11 ⊢ 0 ∈ ℕ0 | |
| 19 | 18 | fconst6 6797 | . . . . . . . . . 10 ⊢ (1o × {0}):1o⟶ℕ0 | 
| 20 | nn0ex 12534 | . . . . . . . . . . 11 ⊢ ℕ0 ∈ V | |
| 21 | 1oex 8517 | . . . . . . . . . . 11 ⊢ 1o ∈ V | |
| 22 | 20, 21 | elmap 8912 | . . . . . . . . . 10 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {0}):1o⟶ℕ0) | 
| 23 | 19, 22 | mpbir 231 | . . . . . . . . 9 ⊢ (1o × {0}) ∈ (ℕ0 ↑m 1o) | 
| 24 | ne0i 4340 | . . . . . . . . 9 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) → (ℕ0 ↑m 1o) ≠ ∅) | |
| 25 | map0b 8924 | . . . . . . . . 9 ⊢ ((ℕ0 ↑m 1o) ≠ ∅ → (∅ ↑m (ℕ0 ↑m 1o)) = ∅) | |
| 26 | 23, 24, 25 | mp2b 10 | . . . . . . . 8 ⊢ (∅ ↑m (ℕ0 ↑m 1o)) = ∅ | 
| 27 | 17, 26 | eqtr2i 2765 | . . . . . . 7 ⊢ ∅ = (Base‘(1o mPwSer ∅)) | 
| 28 | eqid 2736 | . . . . . . 7 ⊢ (+g‘∅) = (+g‘∅) | |
| 29 | eqid 2736 | . . . . . . 7 ⊢ (+g‘(1o mPwSer ∅)) = (+g‘(1o mPwSer ∅)) | |
| 30 | 8, 27, 28, 29 | psrplusg 21957 | . . . . . 6 ⊢ (+g‘(1o mPwSer ∅)) = ( ∘f (+g‘∅) ↾ (∅ × ∅)) | 
| 31 | xp0 6177 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
| 32 | 31 | reseq2i 5993 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ (∅ × ∅)) = ( ∘f (+g‘∅) ↾ ∅) | 
| 33 | 10, 30, 32 | 3eqtri 2768 | . . . . 5 ⊢ (+g‘(1o mPoly ∅)) = ( ∘f (+g‘∅) ↾ ∅) | 
| 34 | res0 6000 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ ∅) = ∅ | |
| 35 | plusgid 17325 | . . . . . . 7 ⊢ +g = Slot (+g‘ndx) | |
| 36 | 35 | str0 17227 | . . . . . 6 ⊢ ∅ = (+g‘∅) | 
| 37 | 34, 36 | eqtri 2764 | . . . . 5 ⊢ ( ∘f (+g‘∅) ↾ ∅) = (+g‘∅) | 
| 38 | 7, 33, 37 | 3eqtri 2768 | . . . 4 ⊢ (+g‘(Poly1‘∅)) = (+g‘∅) | 
| 39 | fvprc 6897 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → ( I ‘𝑅) = ∅) | |
| 40 | 39 | fveq2d 6909 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘∅)) | 
| 41 | 40 | fveq2d 6909 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘∅))) | 
| 42 | fvprc 6897 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
| 43 | 42 | fveq2d 6909 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘𝑅)) = (+g‘∅)) | 
| 44 | 38, 41, 43 | 3eqtr4a 2802 | . . 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 2745 | 1 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘( I ‘𝑅))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∅c0 4332 {csn 4625 I cid 5576 × cxp 5682 ↾ cres 5686 Oncon0 6383 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 1oc1o 8500 ↑m cmap 8867 0cc0 11156 ℕ0cn0 12528 ndxcnx 17231 Basecbs 17248 +gcplusg 17298 mPwSer cmps 21925 mPoly cmpl 21927 Poly1cpl1 22179 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 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 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-tset 17317 df-ple 17318 df-psr 21930 df-mpl 21932 df-opsr 21934 df-psr1 22182 df-ply1 22184 | 
| This theorem is referenced by: (None) | 
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