![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dgrid | Structured version Visualization version GIF version |
Description: The degree of the identity function. (Contributed by Mario Carneiro, 26-Jul-2014.) |
Ref | Expression |
---|---|
dgrid | ⊢ (deg‘Xp) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 11174 | . 2 ⊢ 1 ∈ ℂ | |
2 | ax-1ne0 11185 | . 2 ⊢ 1 ≠ 0 | |
3 | 1nn0 12495 | . 2 ⊢ 1 ∈ ℕ0 | |
4 | mptresid 6050 | . . . 4 ⊢ ( I ↾ ℂ) = (𝑧 ∈ ℂ ↦ 𝑧) | |
5 | df-idp 26041 | . . . 4 ⊢ Xp = ( I ↾ ℂ) | |
6 | exp1 14040 | . . . . . . 7 ⊢ (𝑧 ∈ ℂ → (𝑧↑1) = 𝑧) | |
7 | 6 | oveq2d 7428 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → (1 · (𝑧↑1)) = (1 · 𝑧)) |
8 | mullid 11220 | . . . . . 6 ⊢ (𝑧 ∈ ℂ → (1 · 𝑧) = 𝑧) | |
9 | 7, 8 | eqtrd 2771 | . . . . 5 ⊢ (𝑧 ∈ ℂ → (1 · (𝑧↑1)) = 𝑧) |
10 | 9 | mpteq2ia 5251 | . . . 4 ⊢ (𝑧 ∈ ℂ ↦ (1 · (𝑧↑1))) = (𝑧 ∈ ℂ ↦ 𝑧) |
11 | 4, 5, 10 | 3eqtr4i 2769 | . . 3 ⊢ Xp = (𝑧 ∈ ℂ ↦ (1 · (𝑧↑1))) |
12 | 11 | dgr1term 26112 | . 2 ⊢ ((1 ∈ ℂ ∧ 1 ≠ 0 ∧ 1 ∈ ℕ0) → (deg‘Xp) = 1) |
13 | 1, 2, 3, 12 | mp3an 1460 | 1 ⊢ (deg‘Xp) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ↦ cmpt 5231 I cid 5573 ↾ cres 5678 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 0cc0 11116 1c1 11117 · cmul 11121 ℕ0cn0 12479 ↑cexp 14034 Xpcidp 26037 degcdgr 26039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-sum 15640 df-0p 25519 df-ply 26040 df-idp 26041 df-coe 26042 df-dgr 26043 |
This theorem is referenced by: plyremlem 26156 |
Copyright terms: Public domain | W3C validator |