fldextrspunlem2 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fldextrspunlem2		Structured version Visualization version GIF version

Theorem fldextrspunlem2 34131

Description: Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)

**Hypotheses**
Ref	Expression
fldextrspunfld.k	⊢ 𝐾 = (𝐿 ↾_s 𝐹)
fldextrspunfld.i	⊢ 𝐼 = (𝐿 ↾_s 𝐺)
fldextrspunfld.j	⊢ 𝐽 = (𝐿 ↾_s 𝐻)
fldextrspunfld.2	⊢ (𝜑 → 𝐿 ∈ Field)
fldextrspunfld.3	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
fldextrspunfld.4	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
fldextrspunfld.5	⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
fldextrspunfld.6	⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
fldextrspunfld.7	⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
fldextrspunfld.n	⊢ 𝑁 = (RingSpan‘𝐿)
fldextrspunfld.c	⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))
fldextrspunfld.e	⊢ 𝐸 = (𝐿 ↾_s 𝐶)

**Assertion**
Ref	Expression
fldextrspunlem2	⊢ (𝜑 → 𝐶 = (𝐿 fldGen (𝐺 ∪ 𝐻)))

**Proof of Theorem fldextrspunlem2**
Step	Hyp	Ref	Expression
1		fldextrspunfld.2	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Field)
2	1	flddrngd 20891	. . . 4 ⊢ (𝜑 → 𝐿 ∈ DivRing)
3	2	drngringd 20885	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
4		eqidd 2766	. . 3 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
5		fldextrspunfld.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
6		eqid 2765	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
7	6	sdrgss 20946	. . . . 5 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
8	5, 7	syl 18	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
9		fldextrspunfld.6	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
10	6	sdrgss 20946	. . . . 5 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
12	8, 11	unssd 4145	. . 3 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ (Base‘𝐿))
13		fldextrspunfld.n	. . . 4 ⊢ 𝑁 = (RingSpan‘𝐿)
14	13	a1i 11	. . 3 ⊢ (𝜑 → 𝑁 = (RingSpan‘𝐿))
15		fldextrspunfld.c	. . . 4 ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))
16	15	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 = (𝑁‘(𝐺 ∪ 𝐻)))
17	6, 2, 12	fldgensdrg 33699	. . . 4 ⊢ (𝜑 → (𝐿 fldGen (𝐺 ∪ 𝐻)) ∈ (SubDRing‘𝐿))
18		sdrgsubrg 20944	. . . 4 ⊢ ((𝐿 fldGen (𝐺 ∪ 𝐻)) ∈ (SubDRing‘𝐿) → (𝐿 fldGen (𝐺 ∪ 𝐻)) ∈ (SubRing‘𝐿))
19	17, 18	syl 18	. . 3 ⊢ (𝜑 → (𝐿 fldGen (𝐺 ∪ 𝐻)) ∈ (SubRing‘𝐿))
20	6, 2, 12	fldgenssid 33698	. . 3 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ (𝐿 fldGen (𝐺 ∪ 𝐻)))
21	3, 4, 12, 14, 16, 19, 20	rgspnmin 20764	. 2 ⊢ (𝜑 → 𝐶 ⊆ (𝐿 fldGen (𝐺 ∪ 𝐻)))
22	3, 4, 12, 14, 16	rgspncl 20762	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝐿))
23		fldextrspunfld.e	. . . . 5 ⊢ 𝐸 = (𝐿 ↾_s 𝐶)
24		fldextrspunfld.k	. . . . . . 7 ⊢ 𝐾 = (𝐿 ↾_s 𝐹)
25		fldextrspunfld.i	. . . . . . 7 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
26		fldextrspunfld.j	. . . . . . 7 ⊢ 𝐽 = (𝐿 ↾_s 𝐻)
27		fldextrspunfld.3	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
28		fldextrspunfld.4	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
29		fldextrspunfld.7	. . . . . . 7 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
30	24, 25, 26, 1, 27, 28, 5, 9, 29, 13, 15, 23	fldextrspunfld 34130	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Field)
31	30	flddrngd 20891	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ DivRing)
32	23, 31	eqeltrrid 2870	. . . 4 ⊢ (𝜑 → (𝐿 ↾_s 𝐶) ∈ DivRing)
33		issdrg 20941	. . . 4 ⊢ (𝐶 ∈ (SubDRing‘𝐿) ↔ (𝐿 ∈ DivRing ∧ 𝐶 ∈ (SubRing‘𝐿) ∧ (𝐿 ↾_s 𝐶) ∈ DivRing))
34	2, 22, 32, 33	syl3anbrc 1362	. . 3 ⊢ (𝜑 → 𝐶 ∈ (SubDRing‘𝐿))
35	3, 4, 12, 14, 16	rgspnssid 20763	. . 3 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
36	6, 2, 34, 35	fldgenssp 33703	. 2 ⊢ (𝜑 → (𝐿 fldGen (𝐺 ∪ 𝐻)) ⊆ 𝐶)
37	21, 36	eqssd 3955	1 ⊢ (𝜑 → 𝐶 = (𝐿 fldGen (𝐺 ∪ 𝐻)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ∪ cun 3904 ⊆ wss 3906 ‘cfv 6540 (class class class)co 7419 ℕ₀cn0 12519 Basecbs 17291 ↾_s cress 17312 SubRingcsubrg 20718 RingSpancrgspn 20759 DivRingcdr 20877 Fieldcfield 20878 SubDRingcsdrg 20939 fldGen cfldgen 33695 [:]cextdg 34094

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-reg 9561 ax-inf2 9617 ax-ac2 10462 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-rpss 7730 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-sup 9409 df-inf 9410 df-oi 9479 df-r1 9743 df-rank 9744 df-scott 9865 df-dju 9903 df-card 9941 df-acn 9944 df-ac 10116 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-ind 12234 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-xnn0 12593 df-z 12607 df-dec 12728 df-uz 12879 df-rp 13033 df-xadd 13154 df-fz 13552 df-fzo 13700 df-seq 14056 df-exp 14116 df-hash 14385 df-word 14569 df-lsw 14618 df-concat 14626 df-s1 14653 df-substr 14699 df-pfx 14731 df-s2 14909 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-clim 15563 df-sum 15762 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ocomp 17353 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-0g 17516 df-gsum 17517 df-prds 17522 df-pws 17524 df-mre 17660 df-mrc 17661 df-mri 17662 df-acs 17663 df-proset 18372 df-drs 18373 df-poset 18391 df-ipo 18606 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-mhm 18878 df-submnd 18879 df-grp 19047 df-minusg 19048 df-sbg 19049 df-mulg 19178 df-subg 19233 df-ghm 19328 df-cntz 19431 df-cntr 19432 df-lsm 19750 df-cmn 19896 df-abl 19897 df-mgp 20261 df-rng 20275 df-ur 20308 df-ring 20361 df-cring 20362 df-oppr 20465 df-dvdsr 20485 df-unit 20486 df-invr 20516 df-dvr 20529 df-nzr 20660 df-subrng 20695 df-subrg 20719 df-rgspn 20760 df-rlreg 20843 df-domn 20844 df-idom 20845 df-drng 20879 df-field 20880 df-sdrg 20940 df-lmod 21033 df-lss 21103 df-lsp 21143 df-lmhm 21193 df-lmim 21194 df-lbs 21246 df-lvec 21274 df-sra 21344 df-rgmod 21345 df-cnfld 21573 df-zring 21647 df-dsmm 21932 df-frlm 21947 df-uvc 21983 df-lindf 22006 df-linds 22007 df-assa 22053 df-fldgen 33696 df-dim 34054 df-fldext 34095 df-extdg 34096

This theorem is used by: fldextrspundgle 34132 fldextrspundgdvdslem 34134 fldextrspundgdvds 34135

W3C validator