Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > srgbinomlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for srgbinomlem 19695. (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 480 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝑅 ∈ SRing) |
| 3 | srgbinom.g | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅) | |
| 4 | 3 | srgmgp 19661 | . . . . 5 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
| 5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 6 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐺 ∈ Mnd) |
| 7 | simprl 767 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐷 ∈ ℕ0) | |
| 8 | srgbinomlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
| 9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐴 ∈ 𝑆) |
| 10 | srgbinom.s | . . . . 5 ⊢ 𝑆 = (Base‘𝑅) | |
| 11 | 3, 10 | mgpbas 19641 | . . . 4 ⊢ 𝑆 = (Base‘𝐺) |
| 12 | srgbinom.e | . . . 4 ⊢ ↑ = (.g‘𝐺) | |
| 13 | 11, 12 | mulgnn0cl 18635 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐷 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → (𝐷 ↑ 𝐴) ∈ 𝑆) |
| 14 | 6, 7, 9, 13 | syl3anc 1369 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐷 ↑ 𝐴) ∈ 𝑆) |
| 15 | simprr 769 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐸 ∈ ℕ0) | |
| 16 | srgbinomlem.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
| 17 | 16 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐵 ∈ 𝑆) |
| 18 | 11, 12 | mulgnn0cl 18635 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 ∈ ℕ0 ∧ 𝐵 ∈ 𝑆) → (𝐸 ↑ 𝐵) ∈ 𝑆) |
| 19 | 6, 15, 17, 18 | syl3anc 1369 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐸 ↑ 𝐵) ∈ 𝑆) |
| 20 | srgbinom.m | . . 3 ⊢ × = (.r‘𝑅) | |
| 21 | 10, 20 | srgcl 19663 | . 2 ⊢ ((𝑅 ∈ SRing ∧ (𝐷 ↑ 𝐴) ∈ 𝑆 ∧ (𝐸 ↑ 𝐵) ∈ 𝑆) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
| 22 | 2, 14, 19, 21 | syl3anc 1369 | 1 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 ℕ0cn0 12163 Basecbs 16840 +gcplusg 16888 .rcmulr 16889 Mndcmnd 18300 .gcmg 18615 mulGrpcmgp 19635 SRingcsrg 19656 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-seq 13650 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mulg 18616 df-mgp 19636 df-srg 19657 |
| This theorem is referenced by: srgbinomlem2 19692 srgbinomlem3 19693 |
| Copyright terms: Public domain | W3C validator |