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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ply1moneq | Structured version Visualization version GIF version |
Description: Two monomials are equal iff their powers are equal. (Contributed by Thierry Arnoux, 20-Feb-2025.) |
Ref | Expression |
---|---|
ply1moneq.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1moneq.x | ⊢ 𝑋 = (var1‘𝑅) |
ply1moneq.e | ⊢ ↑ = (.g‘(mulGrp‘𝑃)) |
ply1moneq.r | ⊢ (𝜑 → 𝑅 ∈ NzRing) |
ply1moneq.m | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
ply1moneq.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
ply1moneq | ⊢ (𝜑 → ((𝑀 ↑ 𝑋) = (𝑁 ↑ 𝑋) ↔ 𝑀 = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1moneq.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | ply1moneq.x | . . . . . . 7 ⊢ 𝑋 = (var1‘𝑅) | |
3 | ply1moneq.e | . . . . . . 7 ⊢ ↑ = (.g‘(mulGrp‘𝑃)) | |
4 | ply1moneq.r | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ NzRing) | |
5 | nzrring 20408 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
7 | ply1moneq.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
8 | eqid 2731 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
9 | eqid 2731 | . . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
10 | 1, 2, 3, 6, 7, 8, 9 | coe1mon 32939 | . . . . . 6 ⊢ (𝜑 → (coe1‘(𝑀 ↑ 𝑋)) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝑀, (1r‘𝑅), (0g‘𝑅)))) |
11 | fvexd 6906 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1r‘𝑅) ∈ V) | |
12 | fvexd 6906 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (0g‘𝑅) ∈ V) | |
13 | 11, 12 | ifcld 4574 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 𝑀, (1r‘𝑅), (0g‘𝑅)) ∈ V) |
14 | 10, 13 | fvmpt2d 7011 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑀 ↑ 𝑋))‘𝑘) = if(𝑘 = 𝑀, (1r‘𝑅), (0g‘𝑅))) |
15 | ply1moneq.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
16 | 1, 2, 3, 6, 15, 8, 9 | coe1mon 32939 | . . . . . 6 ⊢ (𝜑 → (coe1‘(𝑁 ↑ 𝑋)) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝑁, (1r‘𝑅), (0g‘𝑅)))) |
17 | 11, 12 | ifcld 4574 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 𝑁, (1r‘𝑅), (0g‘𝑅)) ∈ V) |
18 | 16, 17 | fvmpt2d 7011 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((coe1‘(𝑁 ↑ 𝑋))‘𝑘) = if(𝑘 = 𝑁, (1r‘𝑅), (0g‘𝑅))) |
19 | 14, 18 | eqeq12d 2747 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((coe1‘(𝑀 ↑ 𝑋))‘𝑘) = ((coe1‘(𝑁 ↑ 𝑋))‘𝑘) ↔ if(𝑘 = 𝑀, (1r‘𝑅), (0g‘𝑅)) = if(𝑘 = 𝑁, (1r‘𝑅), (0g‘𝑅)))) |
20 | 9, 8 | nzrnz 20407 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
21 | 4, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅)) |
22 | 21 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1r‘𝑅) ≠ (0g‘𝑅)) |
23 | ifnebib 32049 | . . . . 5 ⊢ ((1r‘𝑅) ≠ (0g‘𝑅) → (if(𝑘 = 𝑀, (1r‘𝑅), (0g‘𝑅)) = if(𝑘 = 𝑁, (1r‘𝑅), (0g‘𝑅)) ↔ (𝑘 = 𝑀 ↔ 𝑘 = 𝑁))) | |
24 | 22, 23 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 𝑀, (1r‘𝑅), (0g‘𝑅)) = if(𝑘 = 𝑁, (1r‘𝑅), (0g‘𝑅)) ↔ (𝑘 = 𝑀 ↔ 𝑘 = 𝑁))) |
25 | 19, 24 | bitrd 279 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((coe1‘(𝑀 ↑ 𝑋))‘𝑘) = ((coe1‘(𝑁 ↑ 𝑋))‘𝑘) ↔ (𝑘 = 𝑀 ↔ 𝑘 = 𝑁))) |
26 | 25 | ralbidva 3174 | . 2 ⊢ (𝜑 → (∀𝑘 ∈ ℕ0 ((coe1‘(𝑀 ↑ 𝑋))‘𝑘) = ((coe1‘(𝑁 ↑ 𝑋))‘𝑘) ↔ ∀𝑘 ∈ ℕ0 (𝑘 = 𝑀 ↔ 𝑘 = 𝑁))) |
27 | eqid 2731 | . . . . 5 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
28 | eqid 2731 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
29 | 1, 2, 27, 3, 28 | ply1moncl 22014 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℕ0) → (𝑀 ↑ 𝑋) ∈ (Base‘𝑃)) |
30 | 6, 7, 29 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑀 ↑ 𝑋) ∈ (Base‘𝑃)) |
31 | 1, 2, 27, 3, 28 | ply1moncl 22014 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℕ0) → (𝑁 ↑ 𝑋) ∈ (Base‘𝑃)) |
32 | 6, 15, 31 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (Base‘𝑃)) |
33 | eqid 2731 | . . . 4 ⊢ (coe1‘(𝑀 ↑ 𝑋)) = (coe1‘(𝑀 ↑ 𝑋)) | |
34 | eqid 2731 | . . . 4 ⊢ (coe1‘(𝑁 ↑ 𝑋)) = (coe1‘(𝑁 ↑ 𝑋)) | |
35 | 1, 28, 33, 34 | ply1coe1eq 22043 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑁 ↑ 𝑋) ∈ (Base‘𝑃)) → (∀𝑘 ∈ ℕ0 ((coe1‘(𝑀 ↑ 𝑋))‘𝑘) = ((coe1‘(𝑁 ↑ 𝑋))‘𝑘) ↔ (𝑀 ↑ 𝑋) = (𝑁 ↑ 𝑋))) |
36 | 6, 30, 32, 35 | syl3anc 1370 | . 2 ⊢ (𝜑 → (∀𝑘 ∈ ℕ0 ((coe1‘(𝑀 ↑ 𝑋))‘𝑘) = ((coe1‘(𝑁 ↑ 𝑋))‘𝑘) ↔ (𝑀 ↑ 𝑋) = (𝑁 ↑ 𝑋))) |
37 | 7, 15 | eqelbid 31983 | . 2 ⊢ (𝜑 → (∀𝑘 ∈ ℕ0 (𝑘 = 𝑀 ↔ 𝑘 = 𝑁) ↔ 𝑀 = 𝑁)) |
38 | 26, 36, 37 | 3bitr3d 309 | 1 ⊢ (𝜑 → ((𝑀 ↑ 𝑋) = (𝑁 ↑ 𝑋) ↔ 𝑀 = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 Vcvv 3473 ifcif 4528 ‘cfv 6543 (class class class)co 7412 ℕ0cn0 12477 Basecbs 17149 0gc0g 17390 .gcmg 18987 mulGrpcmgp 20029 1rcur 20076 Ringcrg 20128 NzRingcnzr 20404 var1cv1 21920 Poly1cpl1 21921 coe1cco1 21922 |
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-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
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-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 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-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-oi 9509 df-card 9938 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-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-fzo 13633 df-seq 13972 df-hash 14296 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-srg 20082 df-ring 20130 df-nzr 20405 df-subrng 20435 df-subrg 20460 df-lmod 20617 df-lss 20688 df-psr 21682 df-mvr 21683 df-mpl 21684 df-opsr 21686 df-psr1 21924 df-vr1 21925 df-ply1 21926 df-coe1 21927 |
This theorem is referenced by: ply1degltdimlem 32996 ply1degltdim 32997 |
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