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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mzpresrename | Structured version Visualization version GIF version |
Description: A polynomial is a polynomial over all larger index sets. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.) |
Ref | Expression |
---|---|
mzpresrename | ⊢ ((𝑊 ∈ V ∧ 𝑉 ⊆ 𝑊 ∧ 𝐹 ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ↾ 𝑉))) ∈ (mzPoly‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coires1 6264 | . . . 4 ⊢ (𝑥 ∘ ( I ↾ 𝑉)) = (𝑥 ↾ 𝑉) | |
2 | 1 | fveq2i 6895 | . . 3 ⊢ (𝐹‘(𝑥 ∘ ( I ↾ 𝑉))) = (𝐹‘(𝑥 ↾ 𝑉)) |
3 | 2 | mpteq2i 5254 | . 2 ⊢ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ∘ ( I ↾ 𝑉)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ↾ 𝑉))) |
4 | simp1 1135 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝑉 ⊆ 𝑊 ∧ 𝐹 ∈ (mzPoly‘𝑉)) → 𝑊 ∈ V) | |
5 | simp3 1137 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝑉 ⊆ 𝑊 ∧ 𝐹 ∈ (mzPoly‘𝑉)) → 𝐹 ∈ (mzPoly‘𝑉)) | |
6 | f1oi 6872 | . . . . . 6 ⊢ ( I ↾ 𝑉):𝑉–1-1-onto→𝑉 | |
7 | f1of 6834 | . . . . . 6 ⊢ (( I ↾ 𝑉):𝑉–1-1-onto→𝑉 → ( I ↾ 𝑉):𝑉⟶𝑉) | |
8 | 6, 7 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝑉):𝑉⟶𝑉 |
9 | fss 6735 | . . . . 5 ⊢ ((( I ↾ 𝑉):𝑉⟶𝑉 ∧ 𝑉 ⊆ 𝑊) → ( I ↾ 𝑉):𝑉⟶𝑊) | |
10 | 8, 9 | mpan 687 | . . . 4 ⊢ (𝑉 ⊆ 𝑊 → ( I ↾ 𝑉):𝑉⟶𝑊) |
11 | 10 | 3ad2ant2 1133 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝑉 ⊆ 𝑊 ∧ 𝐹 ∈ (mzPoly‘𝑉)) → ( I ↾ 𝑉):𝑉⟶𝑊) |
12 | mzprename 41790 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ( I ↾ 𝑉):𝑉⟶𝑊) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ∘ ( I ↾ 𝑉)))) ∈ (mzPoly‘𝑊)) | |
13 | 4, 5, 11, 12 | syl3anc 1370 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝑉 ⊆ 𝑊 ∧ 𝐹 ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ∘ ( I ↾ 𝑉)))) ∈ (mzPoly‘𝑊)) |
14 | 3, 13 | eqeltrrid 2837 | 1 ⊢ ((𝑊 ∈ V ∧ 𝑉 ⊆ 𝑊 ∧ 𝐹 ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑥 ↾ 𝑉))) ∈ (mzPoly‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3949 ↦ cmpt 5232 I cid 5574 ↾ cres 5679 ∘ ccom 5681 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7412 ↑m cmap 8823 ℤcz 12563 mzPolycmzp 41763 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
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-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 df-mzpcl 41764 df-mzp 41765 |
This theorem is referenced by: mzpcompact2lem 41792 diophin 41813 rabdiophlem2 41843 |
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