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Mirrors > Home > MPE Home > Th. List > srgbinomlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for srgbinomlem 19915. (Contributed by AV, 23-Aug-2019.) |
Ref | Expression |
---|---|
srgbinom.s | ⊢ 𝑆 = (Base‘𝑅) |
srgbinom.m | ⊢ × = (.r‘𝑅) |
srgbinom.t | ⊢ · = (.g‘𝑅) |
srgbinom.a | ⊢ + = (+g‘𝑅) |
srgbinom.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
srgbinom.e | ⊢ ↑ = (.g‘𝐺) |
srgbinomlem.r | ⊢ (𝜑 → 𝑅 ∈ SRing) |
srgbinomlem.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
srgbinomlem.b | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
srgbinomlem.c | ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) |
srgbinomlem.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
srgbinomlem2 | ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐶 · ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵))) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgbinom.s | . 2 ⊢ 𝑆 = (Base‘𝑅) | |
2 | srgbinom.t | . 2 ⊢ · = (.g‘𝑅) | |
3 | srgbinomlem.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ SRing) | |
4 | srgmnd 19880 | . . . 4 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
6 | 5 | adantr 482 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝑅 ∈ Mnd) |
7 | simpr1 1195 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐶 ∈ ℕ0) | |
8 | srgbinom.m | . . . 4 ⊢ × = (.r‘𝑅) | |
9 | srgbinom.a | . . . 4 ⊢ + = (+g‘𝑅) | |
10 | srgbinom.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑅) | |
11 | srgbinom.e | . . . 4 ⊢ ↑ = (.g‘𝐺) | |
12 | srgbinomlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
13 | srgbinomlem.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
14 | srgbinomlem.c | . . . 4 ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) | |
15 | srgbinomlem.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
16 | 1, 8, 2, 9, 10, 11, 3, 12, 13, 14, 15 | srgbinomlem1 19911 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
17 | 16 | 3adantr1 1170 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
18 | 1, 2, 6, 7, 17 | mulgnn0cld 18856 | 1 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐶 · ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵))) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6494 (class class class)co 7352 ℕ0cn0 12372 Basecbs 17043 +gcplusg 17093 .rcmulr 17094 Mndcmnd 18516 .gcmg 18831 mulGrpcmgp 19855 SRingcsrg 19876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-n0 12373 df-z 12459 df-uz 12723 df-fz 13380 df-seq 13862 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-plusg 17106 df-0g 17283 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mulg 18832 df-cmn 19523 df-mgp 19856 df-srg 19877 |
This theorem is referenced by: srgbinomlem3 19913 srgbinomlem4 19914 |
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