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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 6948 | . . . . 5 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
2 | 1 | fveq2d 6877 | . . . 4 ⊢ (𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘𝑅)) |
3 | 2 | fveq2d 6877 | . . 3 ⊢ (𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘𝑅))) |
4 | eqid 2731 | . . . . . 6 ⊢ (Poly1‘∅) = (Poly1‘∅) | |
5 | eqid 2731 | . . . . . 6 ⊢ (1o mPoly ∅) = (1o mPoly ∅) | |
6 | eqid 2731 | . . . . . 6 ⊢ (+g‘(Poly1‘∅)) = (+g‘(Poly1‘∅)) | |
7 | 4, 5, 6 | ply1plusg 21671 | . . . . 5 ⊢ (+g‘(Poly1‘∅)) = (+g‘(1o mPoly ∅)) |
8 | eqid 2731 | . . . . . . 7 ⊢ (1o mPwSer ∅) = (1o mPwSer ∅) | |
9 | eqid 2731 | . . . . . . 7 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPoly ∅)) | |
10 | 5, 8, 9 | mplplusg 21666 | . . . . . 6 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPwSer ∅)) |
11 | base0 17126 | . . . . . . . . . 10 ⊢ ∅ = (Base‘∅) | |
12 | psr1baslem 21631 | . . . . . . . . . 10 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
13 | eqid 2731 | . . . . . . . . . 10 ⊢ (Base‘(1o mPwSer ∅)) = (Base‘(1o mPwSer ∅)) | |
14 | 1on 8455 | . . . . . . . . . . 11 ⊢ 1o ∈ On | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 1o ∈ On) |
16 | 8, 11, 12, 13, 15 | psrbas 21421 | . . . . . . . . 9 ⊢ (⊤ → (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o))) |
17 | 16 | mptru 1548 | . . . . . . . 8 ⊢ (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o)) |
18 | 0nn0 12464 | . . . . . . . . . . 11 ⊢ 0 ∈ ℕ0 | |
19 | 18 | fconst6 6763 | . . . . . . . . . 10 ⊢ (1o × {0}):1o⟶ℕ0 |
20 | nn0ex 12455 | . . . . . . . . . . 11 ⊢ ℕ0 ∈ V | |
21 | 1oex 8453 | . . . . . . . . . . 11 ⊢ 1o ∈ V | |
22 | 20, 21 | elmap 8843 | . . . . . . . . . 10 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {0}):1o⟶ℕ0) |
23 | 19, 22 | mpbir 230 | . . . . . . . . 9 ⊢ (1o × {0}) ∈ (ℕ0 ↑m 1o) |
24 | ne0i 4325 | . . . . . . . . 9 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) → (ℕ0 ↑m 1o) ≠ ∅) | |
25 | map0b 8855 | . . . . . . . . 9 ⊢ ((ℕ0 ↑m 1o) ≠ ∅ → (∅ ↑m (ℕ0 ↑m 1o)) = ∅) | |
26 | 23, 24, 25 | mp2b 10 | . . . . . . . 8 ⊢ (∅ ↑m (ℕ0 ↑m 1o)) = ∅ |
27 | 17, 26 | eqtr2i 2760 | . . . . . . 7 ⊢ ∅ = (Base‘(1o mPwSer ∅)) |
28 | eqid 2731 | . . . . . . 7 ⊢ (+g‘∅) = (+g‘∅) | |
29 | eqid 2731 | . . . . . . 7 ⊢ (+g‘(1o mPwSer ∅)) = (+g‘(1o mPwSer ∅)) | |
30 | 8, 27, 28, 29 | psrplusg 21424 | . . . . . 6 ⊢ (+g‘(1o mPwSer ∅)) = ( ∘f (+g‘∅) ↾ (∅ × ∅)) |
31 | xp0 6141 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
32 | 31 | reseq2i 5965 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ (∅ × ∅)) = ( ∘f (+g‘∅) ↾ ∅) |
33 | 10, 30, 32 | 3eqtri 2763 | . . . . 5 ⊢ (+g‘(1o mPoly ∅)) = ( ∘f (+g‘∅) ↾ ∅) |
34 | res0 5972 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ ∅) = ∅ | |
35 | plusgid 17201 | . . . . . . 7 ⊢ +g = Slot (+g‘ndx) | |
36 | 35 | str0 17099 | . . . . . 6 ⊢ ∅ = (+g‘∅) |
37 | 34, 36 | eqtri 2759 | . . . . 5 ⊢ ( ∘f (+g‘∅) ↾ ∅) = (+g‘∅) |
38 | 7, 33, 37 | 3eqtri 2763 | . . . 4 ⊢ (+g‘(Poly1‘∅)) = (+g‘∅) |
39 | fvprc 6865 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → ( I ‘𝑅) = ∅) | |
40 | 39 | fveq2d 6877 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘∅)) |
41 | 40 | fveq2d 6877 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘∅))) |
42 | fvprc 6865 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
43 | 42 | fveq2d 6877 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘𝑅)) = (+g‘∅)) |
44 | 38, 41, 43 | 3eqtr4a 2797 | . . 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 2740 | 1 ⊢ (+g‘(Poly1‘𝑅)) = (+g‘(Poly1‘( I ‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ≠ wne 2939 Vcvv 3469 ∅c0 4313 {csn 4617 I cid 5561 × cxp 5662 ↾ cres 5666 Oncon0 6348 ⟶wf 6523 ‘cfv 6527 (class class class)co 7388 ∘f cof 7646 1oc1o 8436 ↑m cmap 8798 0cc0 11087 ℕ0cn0 12449 ndxcnx 17103 Basecbs 17121 +gcplusg 17174 mPwSer cmps 21381 mPoly cmpl 21383 Poly1cpl1 21623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5273 ax-sep 5287 ax-nul 5294 ax-pow 5351 ax-pr 5415 ax-un 7703 ax-cnex 11143 ax-resscn 11144 ax-1cn 11145 ax-icn 11146 ax-addcl 11147 ax-addrcl 11148 ax-mulcl 11149 ax-mulrcl 11150 ax-mulcom 11151 ax-addass 11152 ax-mulass 11153 ax-distr 11154 ax-i2m1 11155 ax-1ne0 11156 ax-1rid 11157 ax-rnegex 11158 ax-rrecex 11159 ax-cnre 11160 ax-pre-lttri 11161 ax-pre-lttrn 11162 ax-pre-ltadd 11163 ax-pre-mulgt0 11164 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3375 df-rab 3429 df-v 3471 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4314 df-if 4518 df-pw 4593 df-sn 4618 df-pr 4620 df-tp 4622 df-op 4624 df-uni 4897 df-iun 4987 df-br 5137 df-opab 5199 df-mpt 5220 df-tr 5254 df-id 5562 df-eprel 5568 df-po 5576 df-so 5577 df-fr 5619 df-we 5621 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6284 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7344 df-ov 7391 df-oprab 7392 df-mpo 7393 df-of 7648 df-om 7834 df-1st 7952 df-2nd 7953 df-supp 8124 df-frecs 8243 df-wrecs 8274 df-recs 8348 df-rdg 8387 df-1o 8443 df-er 8681 df-map 8800 df-en 8918 df-dom 8919 df-sdom 8920 df-fin 8921 df-fsupp 9340 df-pnf 11227 df-mnf 11228 df-xr 11229 df-ltxr 11230 df-le 11231 df-sub 11423 df-neg 11424 df-nn 12190 df-2 12252 df-3 12253 df-4 12254 df-5 12255 df-6 12256 df-7 12257 df-8 12258 df-9 12259 df-n0 12450 df-z 12536 df-dec 12655 df-uz 12800 df-fz 13462 df-struct 17057 df-sets 17074 df-slot 17092 df-ndx 17104 df-base 17122 df-ress 17151 df-plusg 17187 df-mulr 17188 df-sca 17190 df-vsca 17191 df-tset 17193 df-ple 17194 df-psr 21386 df-mpl 21388 df-opsr 21390 df-psr1 21626 df-ply1 21628 |
This theorem is referenced by: (None) |
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