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| Mirrors > Home > MPE Home > Th. List > srgbinomlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for srgbinomlem 18931. (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 |
|---|---|
| srgbinomlem1 | ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | srgbinomlem.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ SRing) | |
| 2 | 1 | adantr 474 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝑅 ∈ SRing) |
| 3 | srgbinom.g | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅) | |
| 4 | 3 | srgmgp 18897 | . . . . 5 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
| 5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 6 | 5 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐺 ∈ Mnd) |
| 7 | simprl 761 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐷 ∈ ℕ0) | |
| 8 | srgbinomlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
| 9 | 8 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐴 ∈ 𝑆) |
| 10 | srgbinom.s | . . . . 5 ⊢ 𝑆 = (Base‘𝑅) | |
| 11 | 3, 10 | mgpbas 18882 | . . . 4 ⊢ 𝑆 = (Base‘𝐺) |
| 12 | srgbinom.e | . . . 4 ⊢ ↑ = (.g‘𝐺) | |
| 13 | 11, 12 | mulgnn0cl 17944 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐷 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → (𝐷 ↑ 𝐴) ∈ 𝑆) |
| 14 | 6, 7, 9, 13 | syl3anc 1439 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐷 ↑ 𝐴) ∈ 𝑆) |
| 15 | simprr 763 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐸 ∈ ℕ0) | |
| 16 | srgbinomlem.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
| 17 | 16 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐵 ∈ 𝑆) |
| 18 | 11, 12 | mulgnn0cl 17944 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 ∈ ℕ0 ∧ 𝐵 ∈ 𝑆) → (𝐸 ↑ 𝐵) ∈ 𝑆) |
| 19 | 6, 15, 17, 18 | syl3anc 1439 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐸 ↑ 𝐵) ∈ 𝑆) |
| 20 | srgbinom.m | . . 3 ⊢ × = (.r‘𝑅) | |
| 21 | 10, 20 | srgcl 18899 | . 2 ⊢ ((𝑅 ∈ SRing ∧ (𝐷 ↑ 𝐴) ∈ 𝑆 ∧ (𝐸 ↑ 𝐵) ∈ 𝑆) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
| 22 | 2, 14, 19, 21 | syl3anc 1439 | 1 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 (class class class)co 6922 ℕ0cn0 11642 Basecbs 16255 +gcplusg 16338 .rcmulr 16339 Mndcmnd 17680 .gcmg 17927 mulGrpcmgp 18876 SRingcsrg 18892 |
| 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 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
| 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 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-seq 13120 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mulg 17928 df-mgp 18877 df-srg 18893 |
| This theorem is referenced by: srgbinomlem2 18928 srgbinomlem3 18929 |
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