Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > srgbinomlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for srgbinomlem 19223. (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 | srgbinomlem.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ SRing) | |
2 | srgmnd 19188 | . . . 4 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
4 | 3 | adantr 481 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝑅 ∈ Mnd) |
5 | simpr1 1186 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐶 ∈ ℕ0) | |
6 | srgbinom.s | . . . 4 ⊢ 𝑆 = (Base‘𝑅) | |
7 | srgbinom.m | . . . 4 ⊢ × = (.r‘𝑅) | |
8 | srgbinom.t | . . . 4 ⊢ · = (.g‘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 | 6, 7, 8, 9, 10, 11, 1, 12, 13, 14, 15 | srgbinomlem1 19219 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
17 | 16 | 3adantr1 1161 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
18 | 6, 8 | mulgnn0cl 18182 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝐶 ∈ ℕ0 ∧ ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) → (𝐶 · ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵))) ∈ 𝑆) |
19 | 4, 5, 17, 18 | syl3anc 1363 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ℕ0cn0 11885 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Mndcmnd 17899 .gcmg 18162 mulGrpcmgp 19168 SRingcsrg 19184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-seq 13358 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mulg 18163 df-cmn 18837 df-mgp 19169 df-srg 19185 |
This theorem is referenced by: srgbinomlem3 19221 srgbinomlem4 19222 |
Copyright terms: Public domain | W3C validator |