Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > eldioph4i | Structured version Visualization version GIF version |
Description: Forward-only version of eldioph4b 36393. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
Ref | Expression |
---|---|
eldioph4b.a | ⊢ 𝑊 ∈ V |
eldioph4b.b | ⊢ ¬ 𝑊 ∈ Fin |
eldioph4b.c | ⊢ (𝑊 ∩ ℕ) = ∅ |
Ref | Expression |
---|---|
eldioph4i | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))) → {𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} ∈ (Dioph‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 3722 | . . . . . . . . 9 ⊢ (𝑡 = 𝑎 → (𝑡 ∪ 𝑤) = (𝑎 ∪ 𝑤)) | |
2 | 1 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝑡 = 𝑎 → (𝑃‘(𝑡 ∪ 𝑤)) = (𝑃‘(𝑎 ∪ 𝑤))) |
3 | 2 | eqeq1d 2612 | . . . . . . 7 ⊢ (𝑡 = 𝑎 → ((𝑃‘(𝑡 ∪ 𝑤)) = 0 ↔ (𝑃‘(𝑎 ∪ 𝑤)) = 0)) |
4 | 3 | rexbidv 3034 | . . . . . 6 ⊢ (𝑡 = 𝑎 → (∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0 ↔ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑎 ∪ 𝑤)) = 0)) |
5 | uneq2 3723 | . . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (𝑎 ∪ 𝑤) = (𝑎 ∪ 𝑏)) | |
6 | 5 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝑤 = 𝑏 → (𝑃‘(𝑎 ∪ 𝑤)) = (𝑃‘(𝑎 ∪ 𝑏))) |
7 | 6 | eqeq1d 2612 | . . . . . . 7 ⊢ (𝑤 = 𝑏 → ((𝑃‘(𝑎 ∪ 𝑤)) = 0 ↔ (𝑃‘(𝑎 ∪ 𝑏)) = 0)) |
8 | 7 | cbvrexv 3148 | . . . . . 6 ⊢ (∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑎 ∪ 𝑤)) = 0 ↔ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑎 ∪ 𝑏)) = 0) |
9 | 4, 8 | syl6bb 275 | . . . . 5 ⊢ (𝑡 = 𝑎 → (∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0 ↔ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑎 ∪ 𝑏)) = 0)) |
10 | 9 | cbvrabv 3172 | . . . 4 ⊢ {𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} = {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑎 ∪ 𝑏)) = 0} |
11 | fveq1 6102 | . . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑎 ∪ 𝑏)) = (𝑃‘(𝑎 ∪ 𝑏))) | |
12 | 11 | eqeq1d 2612 | . . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘(𝑎 ∪ 𝑏)) = 0 ↔ (𝑃‘(𝑎 ∪ 𝑏)) = 0)) |
13 | 12 | rexbidv 3034 | . . . . . . 7 ⊢ (𝑝 = 𝑃 → (∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘(𝑎 ∪ 𝑏)) = 0 ↔ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑎 ∪ 𝑏)) = 0)) |
14 | 13 | rabbidv 3164 | . . . . . 6 ⊢ (𝑝 = 𝑃 → {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘(𝑎 ∪ 𝑏)) = 0} = {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑎 ∪ 𝑏)) = 0}) |
15 | 14 | eqeq2d 2620 | . . . . 5 ⊢ (𝑝 = 𝑃 → ({𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} = {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘(𝑎 ∪ 𝑏)) = 0} ↔ {𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} = {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑎 ∪ 𝑏)) = 0})) |
16 | 15 | rspcev 3282 | . . . 4 ⊢ ((𝑃 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁))) ∧ {𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} = {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑎 ∪ 𝑏)) = 0}) → ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁))){𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} = {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘(𝑎 ∪ 𝑏)) = 0}) |
17 | 10, 16 | mpan2 703 | . . 3 ⊢ (𝑃 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁))) → ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁))){𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} = {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘(𝑎 ∪ 𝑏)) = 0}) |
18 | 17 | anim2i 591 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))) → (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁))){𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} = {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘(𝑎 ∪ 𝑏)) = 0})) |
19 | eldioph4b.a | . . 3 ⊢ 𝑊 ∈ V | |
20 | eldioph4b.b | . . 3 ⊢ ¬ 𝑊 ∈ Fin | |
21 | eldioph4b.c | . . 3 ⊢ (𝑊 ∩ ℕ) = ∅ | |
22 | 19, 20, 21 | eldioph4b 36393 | . 2 ⊢ ({𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁))){𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} = {𝑎 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘(𝑎 ∪ 𝑏)) = 0})) |
23 | 18, 22 | sylibr 223 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))) → {𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑃‘(𝑡 ∪ 𝑤)) = 0} ∈ (Dioph‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Fincfn 7841 0cc0 9815 1c1 9816 ℕcn 10897 ℕ0cn0 11169 ...cfz 12197 mzPolycmzp 36303 Diophcdioph 36336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-mzpcl 36304 df-mzp 36305 df-dioph 36337 |
This theorem is referenced by: diophren 36395 |
Copyright terms: Public domain | W3C validator |