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Mirrors > Home > MPE Home > Th. List > srgbinomlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for srgbinomlem 19286. (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 19251 | . . . 4 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Mnd) |
4 | 3 | adantr 483 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝑅 ∈ Mnd) |
5 | simpr1 1189 | . 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 19282 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
17 | 16 | 3adantr1 1164 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
18 | 6, 8 | mulgnn0cl 18236 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝐶 ∈ ℕ0 ∧ ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) → (𝐶 · ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵))) ∈ 𝑆) |
19 | 4, 5, 17, 18 | syl3anc 1366 | 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 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 ℕ0cn0 11889 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Mndcmnd 17903 .gcmg 18216 mulGrpcmgp 19231 SRingcsrg 19247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-seq 13362 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mulg 18217 df-cmn 18900 df-mgp 19232 df-srg 19248 |
This theorem is referenced by: srgbinomlem3 19284 srgbinomlem4 19285 |
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