![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6950 | . . . . 5 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅) | |
2 | 1 | fveq2d 6879 | . . . 4 ⊢ (𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘𝑅)) |
3 | 2 | fveq2d 6879 | . . 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 21673 | . . . . 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 21668 | . . . . . 6 ⊢ (+g‘(1o mPoly ∅)) = (+g‘(1o mPwSer ∅)) |
11 | base0 17128 | . . . . . . . . . 10 ⊢ ∅ = (Base‘∅) | |
12 | psr1baslem 21633 | . . . . . . . . . 10 ⊢ (ℕ0 ↑m 1o) = {𝑎 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
13 | eqid 2731 | . . . . . . . . . 10 ⊢ (Base‘(1o mPwSer ∅)) = (Base‘(1o mPwSer ∅)) | |
14 | 1on 8457 | . . . . . . . . . . 11 ⊢ 1o ∈ On | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 1o ∈ On) |
16 | 8, 11, 12, 13, 15 | psrbas 21423 | . . . . . . . . 9 ⊢ (⊤ → (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o))) |
17 | 16 | mptru 1548 | . . . . . . . 8 ⊢ (Base‘(1o mPwSer ∅)) = (∅ ↑m (ℕ0 ↑m 1o)) |
18 | 0nn0 12466 | . . . . . . . . . . 11 ⊢ 0 ∈ ℕ0 | |
19 | 18 | fconst6 6765 | . . . . . . . . . 10 ⊢ (1o × {0}):1o⟶ℕ0 |
20 | nn0ex 12457 | . . . . . . . . . . 11 ⊢ ℕ0 ∈ V | |
21 | 1oex 8455 | . . . . . . . . . . 11 ⊢ 1o ∈ V | |
22 | 20, 21 | elmap 8845 | . . . . . . . . . 10 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) ↔ (1o × {0}):1o⟶ℕ0) |
23 | 19, 22 | mpbir 230 | . . . . . . . . 9 ⊢ (1o × {0}) ∈ (ℕ0 ↑m 1o) |
24 | ne0i 4327 | . . . . . . . . 9 ⊢ ((1o × {0}) ∈ (ℕ0 ↑m 1o) → (ℕ0 ↑m 1o) ≠ ∅) | |
25 | map0b 8857 | . . . . . . . . 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 21426 | . . . . . 6 ⊢ (+g‘(1o mPwSer ∅)) = ( ∘f (+g‘∅) ↾ (∅ × ∅)) |
31 | xp0 6143 | . . . . . . 7 ⊢ (∅ × ∅) = ∅ | |
32 | 31 | reseq2i 5967 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ (∅ × ∅)) = ( ∘f (+g‘∅) ↾ ∅) |
33 | 10, 30, 32 | 3eqtri 2763 | . . . . 5 ⊢ (+g‘(1o mPoly ∅)) = ( ∘f (+g‘∅) ↾ ∅) |
34 | res0 5974 | . . . . . 6 ⊢ ( ∘f (+g‘∅) ↾ ∅) = ∅ | |
35 | plusgid 17203 | . . . . . . 7 ⊢ +g = Slot (+g‘ndx) | |
36 | 35 | str0 17101 | . . . . . 6 ⊢ ∅ = (+g‘∅) |
37 | 34, 36 | eqtri 2759 | . . . . 5 ⊢ ( ∘f (+g‘∅) ↾ ∅) = (+g‘∅) |
38 | 7, 33, 37 | 3eqtri 2763 | . . . 4 ⊢ (+g‘(Poly1‘∅)) = (+g‘∅) |
39 | fvprc 6867 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → ( I ‘𝑅) = ∅) | |
40 | 39 | fveq2d 6879 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘( I ‘𝑅)) = (Poly1‘∅)) |
41 | 40 | fveq2d 6879 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (+g‘(Poly1‘( I ‘𝑅))) = (+g‘(Poly1‘∅))) |
42 | fvprc 6867 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
43 | 42 | fveq2d 6879 | . . . 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 3470 ∅c0 4315 {csn 4619 I cid 5563 × cxp 5664 ↾ cres 5668 Oncon0 6350 ⟶wf 6525 ‘cfv 6529 (class class class)co 7390 ∘f cof 7648 1oc1o 8438 ↑m cmap 8800 0cc0 11089 ℕ0cn0 12451 ndxcnx 17105 Basecbs 17123 +gcplusg 17176 mPwSer cmps 21383 mPoly cmpl 21385 Poly1cpl1 21625 |
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 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 |
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 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-tp 4624 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7650 df-om 7836 df-1st 7954 df-2nd 7955 df-supp 8126 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-er 8683 df-map 8802 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-fsupp 9342 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-7 12259 df-8 12260 df-9 12261 df-n0 12452 df-z 12538 df-dec 12657 df-uz 12802 df-fz 13464 df-struct 17059 df-sets 17076 df-slot 17094 df-ndx 17106 df-base 17124 df-ress 17153 df-plusg 17189 df-mulr 17190 df-sca 17192 df-vsca 17193 df-tset 17195 df-ple 17196 df-psr 21388 df-mpl 21390 df-opsr 21392 df-psr1 21628 df-ply1 21630 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |