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 33935

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 20778	. . . 4 ⊢ (𝜑 → 𝐿 ∈ DivRing)
3	2	drngringd 20774	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
4		eqidd 2762	. . 3 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
5		fldextrspunfld.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
6		eqid 2761	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
7	6	sdrgss 20830	. . . . 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 20830	. . . . 5 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
12	8, 11	unssd 4142	. . 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 33462	. . . 4 ⊢ (𝜑 → (𝐿 fldGen (𝐺 ∪ 𝐻)) ∈ (SubDRing‘𝐿))
18		sdrgsubrg 20828	. . . 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 33461	. . 3 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ (𝐿 fldGen (𝐺 ∪ 𝐻)))
21	3, 4, 12, 14, 16, 19, 20	rgspnmin 20652	. 2 ⊢ (𝜑 → 𝐶 ⊆ (𝐿 fldGen (𝐺 ∪ 𝐻)))
22	3, 4, 12, 14, 16	rgspncl 20650	. . . 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 33934	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Field)
31	30	flddrngd 20778	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ DivRing)
32	23, 31	eqeltrrid 2866	. . . 4 ⊢ (𝜑 → (𝐿 ↾_s 𝐶) ∈ DivRing)
33		issdrg 20825	. . . 4 ⊢ (𝐶 ∈ (SubDRing‘𝐿) ↔ (𝐿 ∈ DivRing ∧ 𝐶 ∈ (SubRing‘𝐿) ∧ (𝐿 ↾_s 𝐶) ∈ DivRing))
34	2, 22, 32, 33	syl3anbrc 1356	. . 3 ⊢ (𝜑 → 𝐶 ∈ (SubDRing‘𝐿))
35	3, 4, 12, 14, 16	rgspnssid 20651	. . 3 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
36	6, 2, 34, 35	fldgenssp 33466	. 2 ⊢ (𝜑 → (𝐿 fldGen (𝐺 ∪ 𝐻)) ⊆ 𝐶)
37	21, 36	eqssd 3951	1 ⊢ (𝜑 → 𝐶 = (𝐿 fldGen (𝐺 ∪ 𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 ∪ cun 3900 ⊆ wss 3902 ‘cfv 6516 (class class class)co 7391 ℕ₀cn0 12475 Basecbs 17236 ↾_s cress 17257 SubRingcsubrg 20606 RingSpancrgspn 20647 DivRingcdr 20766 Fieldcfield 20767 SubDRingcsdrg 20823 fldGen cfldgen 33458 [:]cextdg 33898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-reg 9534 ax-inf2 9590 ax-ac2 10414 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 ax-addf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-rpss 7701 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-tpos 8200 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-oadd 8435 df-er 8672 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-sup 9382 df-inf 9383 df-oi 9452 df-r1 9716 df-rank 9717 df-dju 9853 df-card 9891 df-acn 9894 df-ac 10066 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-ind 12190 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-xnn0 12549 df-z 12563 df-dec 12683 df-uz 12834 df-rp 12988 df-xadd 13109 df-fz 13507 df-fzo 13654 df-seq 14009 df-exp 14069 df-hash 14338 df-word 14521 df-lsw 14570 df-concat 14578 df-s1 14604 df-substr 14649 df-pfx 14679 df-s2 14855 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-clim 15506 df-sum 15705 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ocomp 17298 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-0g 17461 df-gsum 17462 df-prds 17467 df-pws 17469 df-mre 17605 df-mrc 17606 df-mri 17607 df-acs 17608 df-proset 18317 df-drs 18318 df-poset 18336 df-ipo 18551 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18969 df-minusg 18970 df-sbg 18971 df-mulg 19101 df-subg 19156 df-ghm 19245 df-cntz 19348 df-cntr 19349 df-lsm 19667 df-cmn 19813 df-abl 19814 df-mgp 20178 df-rng 20190 df-ur 20219 df-ring 20272 df-cring 20273 df-oppr 20373 df-dvdsr 20393 df-unit 20394 df-invr 20424 df-dvr 20437 df-nzr 20550 df-subrng 20583 df-subrg 20607 df-rgspn 20648 df-rlreg 20731 df-domn 20732 df-idom 20733 df-drng 20768 df-field 20769 df-sdrg 20824 df-lmod 20917 df-lss 20987 df-lsp 21027 df-lmhm 21077 df-lmim 21078 df-lbs 21130 df-lvec 21158 df-sra 21228 df-rgmod 21229 df-cnfld 21413 df-zring 21487 df-dsmm 21772 df-frlm 21787 df-uvc 21823 df-lindf 21846 df-linds 21847 df-assa 21893 df-fldgen 33459 df-dim 33858 df-fldext 33899 df-extdg 33900

This theorem is referenced by: fldextrspundgle 33936 fldextrspundgdvdslem 33938 fldextrspundgdvds 33939

W3C validator