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Mirrors > Home > MPE Home > Th. List > cnsubrglem | Structured version Visualization version GIF version |
Description: Lemma for resubdrg 21643 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) Avoid ax-mulf 11232. (Revised by GG, 30-Apr-2025.) |
Ref | Expression |
---|---|
cnsubglem.1 | ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) |
cnsubglem.2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) |
cnsubglem.3 | ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴) |
cnsubrglem.4 | ⊢ 1 ∈ 𝐴 |
cnsubrglem.5 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) |
Ref | Expression |
---|---|
cnsubrglem | ⊢ 𝐴 ∈ (SubRing‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnsubglem.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) | |
2 | cnsubglem.2 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) | |
3 | cnsubglem.3 | . . 3 ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴) | |
4 | cnsubrglem.4 | . . 3 ⊢ 1 ∈ 𝐴 | |
5 | 1, 2, 3, 4 | cnsubglem 21450 | . 2 ⊢ 𝐴 ∈ (SubGrp‘ℂfld) |
6 | cnsubrglem.5 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) | |
7 | 1 | adantr 480 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ ℂ) |
8 | 1 | ax-gen 1791 | . . . . . . . . . 10 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) |
9 | eleq1 2826 | . . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
10 | eleq1 2826 | . . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ℂ ↔ 𝑦 ∈ ℂ)) | |
11 | 9, 10 | imbi12d 344 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) ↔ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℂ))) |
12 | 11 | spvv 1993 | . . . . . . . . . 10 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℂ)) |
13 | 8, 12 | ax-mp 5 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℂ) |
14 | 13 | adantl 481 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ) |
15 | 7, 14 | jca 511 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) |
16 | ovmpot 7593 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) = (𝑥 · 𝑦)) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) = (𝑥 · 𝑦)) |
18 | 17 | eqcomd 2740 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) = (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦)) |
19 | 18 | eleq1d 2823 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) ∈ 𝐴)) |
20 | 6, 19 | mpbid 232 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) ∈ 𝐴) |
21 | 20 | rgen2 3196 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) ∈ 𝐴 |
22 | cnring 21420 | . . 3 ⊢ ℂfld ∈ Ring | |
23 | cnfldbas 21385 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
24 | cnfld1 21423 | . . . 4 ⊢ 1 = (1r‘ℂfld) | |
25 | mpocnfldmul 21388 | . . . 4 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (.r‘ℂfld) | |
26 | 23, 24, 25 | issubrg2 20608 | . . 3 ⊢ (ℂfld ∈ Ring → (𝐴 ∈ (SubRing‘ℂfld) ↔ (𝐴 ∈ (SubGrp‘ℂfld) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) ∈ 𝐴))) |
27 | 22, 26 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ (SubRing‘ℂfld) ↔ (𝐴 ∈ (SubGrp‘ℂfld) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑦) ∈ 𝐴)) |
28 | 5, 4, 21, 27 | mpbir3an 1340 | 1 ⊢ 𝐴 ∈ (SubRing‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∀wal 1534 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ℂcc 11150 1c1 11153 + caddc 11155 · cmul 11157 -cneg 11490 SubGrpcsubg 19150 Ringcrg 20250 SubRingcsubrg 20585 ℂfldccnfld 21381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-subrng 20562 df-subrg 20586 df-cnfld 21382 |
This theorem is referenced by: cnsubdrglem 21453 zsubrg 21455 gzsubrg 21456 cnstrcvs 25187 cncvs 25191 |
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