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Mirrors > Home > MPE Home > Th. List > srgbinomlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for srgbinomlem 20135. (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 | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑅) | |
4 | srgbinom.s | . . . 4 ⊢ 𝑆 = (Base‘𝑅) | |
5 | 3, 4 | mgpbas 20045 | . . 3 ⊢ 𝑆 = (Base‘𝐺) |
6 | srgbinom.e | . . 3 ⊢ ↑ = (.g‘𝐺) | |
7 | 3 | srgmgp 20096 | . . . . 5 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐺 ∈ Mnd) |
10 | simprl 768 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐷 ∈ ℕ0) | |
11 | srgbinomlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
12 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐴 ∈ 𝑆) |
13 | 5, 6, 9, 10, 12 | mulgnn0cld 19022 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐷 ↑ 𝐴) ∈ 𝑆) |
14 | simprr 770 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐸 ∈ ℕ0) | |
15 | srgbinomlem.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
16 | 15 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → 𝐵 ∈ 𝑆) |
17 | 5, 6, 9, 14, 16 | mulgnn0cld 19022 | . 2 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐸 ↑ 𝐵) ∈ 𝑆) |
18 | srgbinom.m | . . 3 ⊢ × = (.r‘𝑅) | |
19 | 4, 18 | srgcl 20098 | . 2 ⊢ ((𝑅 ∈ SRing ∧ (𝐷 ↑ 𝐴) ∈ 𝑆 ∧ (𝐸 ↑ 𝐵) ∈ 𝑆) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
20 | 2, 13, 17, 19 | syl3anc 1368 | 1 ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6537 (class class class)co 7405 ℕ0cn0 12476 Basecbs 17153 +gcplusg 17206 .rcmulr 17207 Mndcmnd 18667 .gcmg 18995 mulGrpcmgp 20039 SRingcsrg 20091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-seq 13973 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mulg 18996 df-mgp 20040 df-srg 20092 |
This theorem is referenced by: srgbinomlem2 20132 srgbinomlem3 20133 |
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