fldextrspunlem2 - Mathbox for Thierry Arnoux

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Theorem fldextrspunlem2 33663

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 20645	. . . 4 ⊢ (𝜑 → 𝐿 ∈ DivRing)
3	2	drngringd 20641	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
4		eqidd 2730	. . 3 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
5		fldextrspunfld.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
6		eqid 2729	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
7	6	sdrgss 20697	. . . . 5 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
8	5, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
9		fldextrspunfld.6	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
10	6	sdrgss 20697	. . . . 5 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
11	9, 10	syl 17	. . . 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 33272	. . . 4 ⊢ (𝜑 → (𝐿 fldGen (𝐺 ∪ 𝐻)) ∈ (SubDRing‘𝐿))
18		sdrgsubrg 20695	. . . 4 ⊢ ((𝐿 fldGen (𝐺 ∪ 𝐻)) ∈ (SubDRing‘𝐿) → (𝐿 fldGen (𝐺 ∪ 𝐻)) ∈ (SubRing‘𝐿))
19	17, 18	syl 17	. . 3 ⊢ (𝜑 → (𝐿 fldGen (𝐺 ∪ 𝐻)) ∈ (SubRing‘𝐿))
20	6, 2, 12	fldgenssid 33271	. . 3 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ (𝐿 fldGen (𝐺 ∪ 𝐻)))
21	3, 4, 12, 14, 16, 19, 20	rgspnmin 20519	. 2 ⊢ (𝜑 → 𝐶 ⊆ (𝐿 fldGen (𝐺 ∪ 𝐻)))
22	3, 4, 12, 14, 16	rgspncl 20517	. . . 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 33662	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Field)
31	30	flddrngd 20645	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ DivRing)
32	23, 31	eqeltrrid 2833	. . . 4 ⊢ (𝜑 → (𝐿 ↾_s 𝐶) ∈ DivRing)
33		issdrg 20692	. . . 4 ⊢ (𝐶 ∈ (SubDRing‘𝐿) ↔ (𝐿 ∈ DivRing ∧ 𝐶 ∈ (SubRing‘𝐿) ∧ (𝐿 ↾_s 𝐶) ∈ DivRing))
34	2, 22, 32, 33	syl3anbrc 1344	. . 3 ⊢ (𝜑 → 𝐶 ∈ (SubDRing‘𝐿))
35	3, 4, 12, 14, 16	rgspnssid 20518	. . 3 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
36	6, 2, 34, 35	fldgenssp 33276	. 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

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∪ cun 3903 ⊆ wss 3905 ‘cfv 6486 (class class class)co 7353 ℕ₀cn0 12403 Basecbs 17139 ↾_s cress 17160 SubRingcsubrg 20473 RingSpancrgspn 20514 DivRingcdr 20633 Fieldcfield 20634 SubDRingcsdrg 20690 fldGen cfldgen 33268 [:]cextdg 33626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-reg 9503 ax-inf2 9556 ax-ac2 10376 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-rpss 7663 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-sup 9351 df-inf 9352 df-oi 9421 df-r1 9679 df-rank 9680 df-dju 9816 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-xnn0 12477 df-z 12491 df-dec 12611 df-uz 12755 df-rp 12913 df-xadd 13034 df-fz 13430 df-fzo 13577 df-seq 13928 df-exp 13988 df-hash 14257 df-word 14440 df-lsw 14489 df-concat 14497 df-s1 14522 df-substr 14567 df-pfx 14597 df-s2 14774 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-clim 15414 df-sum 15613 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-starv 17195 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ocomp 17201 df-ds 17202 df-unif 17203 df-hom 17204 df-cco 17205 df-0g 17364 df-gsum 17365 df-prds 17370 df-pws 17372 df-mre 17507 df-mrc 17508 df-mri 17509 df-acs 17510 df-proset 18219 df-drs 18220 df-poset 18238 df-ipo 18453 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-mhm 18676 df-submnd 18677 df-grp 18834 df-minusg 18835 df-sbg 18836 df-mulg 18966 df-subg 19021 df-ghm 19111 df-cntz 19215 df-cntr 19216 df-lsm 19534 df-cmn 19680 df-abl 19681 df-mgp 20045 df-rng 20057 df-ur 20086 df-ring 20139 df-cring 20140 df-oppr 20241 df-dvdsr 20261 df-unit 20262 df-invr 20292 df-dvr 20305 df-nzr 20417 df-subrng 20450 df-subrg 20474 df-rgspn 20515 df-rlreg 20598 df-domn 20599 df-idom 20600 df-drng 20635 df-field 20636 df-sdrg 20691 df-lmod 20784 df-lss 20854 df-lsp 20894 df-lmhm 20945 df-lmim 20946 df-lbs 20998 df-lvec 21026 df-sra 21096 df-rgmod 21097 df-cnfld 21281 df-zring 21373 df-dsmm 21658 df-frlm 21673 df-uvc 21709 df-lindf 21732 df-linds 21733 df-assa 21779 df-ind 32813 df-fldgen 33269 df-dim 33585 df-fldext 33627 df-extdg 33628

This theorem is referenced by: fldextrspundgle 33664 fldextrspundgdvdslem 33666 fldextrspundgdvds 33667

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