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| Mirrors > Home > MPE Home > Th. List > srgbinomlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for srgbinomlem 18935. (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 18900 | . . . 4 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
| 3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
| 4 | 3 | adantr 474 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝑅 ∈ Mnd) |
| 5 | simpr1 1205 | . 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 18931 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
| 17 | 16 | 3adantr1 1171 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
| 18 | 6, 8 | mulgnn0cl 17948 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝐶 ∈ ℕ0 ∧ ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) → (𝐶 · ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵))) ∈ 𝑆) |
| 19 | 4, 5, 17, 18 | syl3anc 1439 | 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 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 ℕ0cn0 11646 Basecbs 16259 +gcplusg 16342 .rcmulr 16343 Mndcmnd 17684 .gcmg 17931 mulGrpcmgp 18880 SRingcsrg 18896 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
| This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-n0 11647 df-z 11733 df-uz 11997 df-fz 12648 df-seq 13124 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-plusg 16355 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-mulg 17932 df-cmn 18585 df-mgp 18881 df-srg 18897 |
| This theorem is referenced by: srgbinomlem3 18933 srgbinomlem4 18934 |
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