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| Mirrors > Home > MPE Home > Th. List > srgbinomlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for srgbinomlem 19780. (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 481 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝑅 ∈ SRing) |
| 3 | srgbinom.g | . . . . . 6 ⊢ 𝐺 = (mulGrp‘𝑅) | |
| 4 | 3 | srgmgp 19746 | . . . . 5 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
| 5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 6 | 5 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐺 ∈ Mnd) |
| 7 | simprl 768 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐷 ∈ ℕ0) | |
| 8 | srgbinomlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
| 9 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐴 ∈ 𝑆) |
| 10 | srgbinom.s | . . . . 5 ⊢ 𝑆 = (Base‘𝑅) | |
| 11 | 3, 10 | mgpbas 19726 | . . . 4 ⊢ 𝑆 = (Base‘𝐺) |
| 12 | srgbinom.e | . . . 4 ⊢ ↑ = (.g‘𝐺) | |
| 13 | 11, 12 | mulgnn0cl 18720 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐷 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → (𝐷 ↑ 𝐴) ∈ 𝑆) |
| 14 | 6, 7, 9, 13 | syl3anc 1370 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐷 ↑ 𝐴) ∈ 𝑆) |
| 15 | simprr 770 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐸 ∈ ℕ0) | |
| 16 | srgbinomlem.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
| 17 | 16 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐵 ∈ 𝑆) |
| 18 | 11, 12 | mulgnn0cl 18720 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐸 ∈ ℕ0 ∧ 𝐵 ∈ 𝑆) → (𝐸 ↑ 𝐵) ∈ 𝑆) |
| 19 | 6, 15, 17, 18 | syl3anc 1370 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐸 ↑ 𝐵) ∈ 𝑆) |
| 20 | srgbinom.m | . . 3 ⊢ × = (.r‘𝑅) | |
| 21 | 10, 20 | srgcl 19748 | . 2 ⊢ ((𝑅 ∈ SRing ∧ (𝐷 ↑ 𝐴) ∈ 𝑆 ∧ (𝐸 ↑ 𝐵) ∈ 𝑆) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
| 22 | 2, 14, 19, 21 | syl3anc 1370 | 1 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 ℕ0cn0 12233 Basecbs 16912 +gcplusg 16962 .rcmulr 16963 Mndcmnd 18385 .gcmg 18700 mulGrpcmgp 19720 SRingcsrg 19741 |
| 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-seq 13722 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mulg 18701 df-mgp 19721 df-srg 19742 |
| This theorem is referenced by: srgbinomlem2 19777 srgbinomlem3 19778 |
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