fldextrspunlem1 - Mathbox for Thierry Arnoux

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Theorem fldextrspunlem1 33676

Description: Lemma for fldextrspunfld 33677. 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
fldextrspunlem1	⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))

Proof of Theorem fldextrspunlem1 Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fldextrspunfld.6	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
2		fldextrspunfld.j	. . . . . 6 ⊢ 𝐽 = (𝐿 ↾_s 𝐻)
3	2	sdrgdrng 20705	. . . . 5 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐽 ∈ DivRing)
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → 𝐽 ∈ DivRing)
5		fldextrspunfld.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
6		eqid 2730	. . . . . 6 ⊢ (𝐽 ↾_s 𝐹) = (𝐽 ↾_s 𝐹)
7	6	sdrgdrng 20705	. . . . 5 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → (𝐽 ↾_s 𝐹) ∈ DivRing)
8	5, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) ∈ DivRing)
9		sdrgsubrg 20706	. . . . . 6 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ∈ (SubRing‘𝐿))
10	1, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (SubRing‘𝐿))
11		fldextrspunfld.5	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
12		sdrgsubrg 20706	. . . . . . . 8 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
13	11, 12	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
14		fldextrspunfld.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
15		sdrgsubrg 20706	. . . . . . . 8 ⊢ (𝐹 ∈ (SubDRing‘𝐼) → 𝐹 ∈ (SubRing‘𝐼))
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐼))
17		fldextrspunfld.i	. . . . . . . . 9 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
18	17	subsubrg 20513	. . . . . . . 8 ⊢ (𝐺 ∈ (SubRing‘𝐿) → (𝐹 ∈ (SubRing‘𝐼) ↔ (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐺)))
19	18	biimpa 476	. . . . . . 7 ⊢ ((𝐺 ∈ (SubRing‘𝐿) ∧ 𝐹 ∈ (SubRing‘𝐼)) → (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐺))
20	13, 16, 19	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐺))
21	20	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿))
22		eqid 2730	. . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘𝐽)
23	22	sdrgss 20708	. . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
24	5, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
25		eqid 2730	. . . . . . . . 9 ⊢ (Base‘𝐿) = (Base‘𝐿)
26	25	sdrgss 20708	. . . . . . . 8 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
27	1, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
28	2, 25	ressbas2 17214	. . . . . . 7 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
29	27, 28	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
30	24, 29	sseqtrrd 3986	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ 𝐻)
31	2	subsubrg 20513	. . . . . 6 ⊢ (𝐻 ∈ (SubRing‘𝐿) → (𝐹 ∈ (SubRing‘𝐽) ↔ (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐻)))
32	31	biimpar 477	. . . . 5 ⊢ ((𝐻 ∈ (SubRing‘𝐿) ∧ (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐻)) → 𝐹 ∈ (SubRing‘𝐽))
33	10, 21, 30, 32	syl12anc 836	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐽))
34		eqid 2730	. . . . 5 ⊢ ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹)
35	34, 6	sralvec 33587	. . . 4 ⊢ ((𝐽 ∈ DivRing ∧ (𝐽 ↾_s 𝐹) ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐽)) → ((subringAlg ‘𝐽)‘𝐹) ∈ LVec)
36	4, 8, 33, 35	syl3anc 1373	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) ∈ LVec)
37		eqid 2730	. . . 4 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
38	37	lbsex 21081	. . 3 ⊢ (((subringAlg ‘𝐽)‘𝐹) ∈ LVec → (LBasis‘((subringAlg ‘𝐽)‘𝐹)) ≠ ∅)
39	36, 38	syl 17	. 2 ⊢ (𝜑 → (LBasis‘((subringAlg ‘𝐽)‘𝐹)) ≠ ∅)
40		fldextrspunfld.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Field)
41		fldidom 20686	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Field → 𝐿 ∈ IDomn)
42	40, 41	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ IDomn)
43	42	idomringd 20643	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ Ring)
44		eqidd 2731	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
45	25	sdrgss 20708	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
46	11, 45	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
47	46, 27	unssd 4157	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ (Base‘𝐿))
48		fldextrspunfld.n	. . . . . . . . . . 11 ⊢ 𝑁 = (RingSpan‘𝐿)
49	48	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 = (RingSpan‘𝐿))
50		fldextrspunfld.c	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))
51	50	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = (𝑁‘(𝐺 ∪ 𝐻)))
52	43, 44, 47, 49, 51	rgspncl 20528	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝐿))
53	43, 44, 47, 49, 51	rgspnssid 20529	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
54	53	unssad 4158	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
55		fldextrspunfld.e	. . . . . . . . . . 11 ⊢ 𝐸 = (𝐿 ↾_s 𝐶)
56	55	subsubrg 20513	. . . . . . . . . 10 ⊢ (𝐶 ∈ (SubRing‘𝐿) → (𝐺 ∈ (SubRing‘𝐸) ↔ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)))
57	56	biimpar 477	. . . . . . . . 9 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)) → 𝐺 ∈ (SubRing‘𝐸))
58	52, 13, 54, 57	syl12anc 836	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐸))
59		eqid 2730	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺)
60	59	sralmod 21100	. . . . . . . 8 ⊢ (𝐺 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
61	58, 60	syl 17	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
62		ressabs 17224	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶) → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
63	52, 54, 62	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
64	55	oveq1i 7399	. . . . . . . . . 10 ⊢ (𝐸 ↾_s 𝐺) = ((𝐿 ↾_s 𝐶) ↾_s 𝐺)
65	63, 64, 17	3eqtr4g 2790	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = 𝐼)
66		eqidd 2731	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
67	25	subrgss 20487	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (SubRing‘𝐿) → 𝐶 ⊆ (Base‘𝐿))
68	52, 67	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐿))
69	55, 25	ressbas2 17214	. . . . . . . . . . . . 13 ⊢ (𝐶 ⊆ (Base‘𝐿) → 𝐶 = (Base‘𝐸))
70	68, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 = (Base‘𝐸))
71	53, 70	sseqtrd 3985	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ (Base‘𝐸))
72	71	unssad 4158	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐸))
73	66, 72	srasca 21093	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
74	65, 73	eqtr3d 2767	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
75	17	sdrgdrng 20705	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐼 ∈ DivRing)
76	11, 75	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ DivRing)
77	74, 76	eqeltrrd 2830	. . . . . . 7 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing)
78		eqid 2730	. . . . . . . 8 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐺)) = (Scalar‘((subringAlg ‘𝐸)‘𝐺))
79	78	islvec 21017	. . . . . . 7 ⊢ (((subringAlg ‘𝐸)‘𝐺) ∈ LVec ↔ (((subringAlg ‘𝐸)‘𝐺) ∈ LMod ∧ (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing))
80	61, 77, 79	sylanbrc 583	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LVec)
81		eqid 2730	. . . . . . 7 ⊢ (LBasis‘((subringAlg ‘𝐸)‘𝐺)) = (LBasis‘((subringAlg ‘𝐸)‘𝐺))
82	81	lbsex 21081	. . . . . 6 ⊢ (((subringAlg ‘𝐸)‘𝐺) ∈ LVec → (LBasis‘((subringAlg ‘𝐸)‘𝐺)) ≠ ∅)
83	80, 82	syl 17	. . . . 5 ⊢ (𝜑 → (LBasis‘((subringAlg ‘𝐸)‘𝐺)) ≠ ∅)
84	83	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (LBasis‘((subringAlg ‘𝐸)‘𝐺)) ≠ ∅)
85	80	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → ((subringAlg ‘𝐸)‘𝐺) ∈ LVec)
86		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺)))
87	81	dimval 33602	. . . . . 6 ⊢ ((((subringAlg ‘𝐸)‘𝐺) ∈ LVec ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) = (♯‘𝑐))
88	85, 86, 87	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) = (♯‘𝑐))
89		eqid 2730	. . . . . 6 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐺)) = (Base‘((subringAlg ‘𝐸)‘𝐺))
90		eqid 2730	. . . . . 6 ⊢ (LSpan‘((subringAlg ‘𝐸)‘𝐺)) = (LSpan‘((subringAlg ‘𝐸)‘𝐺))
91		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
92	91, 37	lbsss 20990	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
93	92	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
94		eqidd 2731	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
95	94, 24	srabase 21090	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
96	29, 95	eqtrd 2765	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
97	96	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
98	93, 97	sseqtrrd 3986	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ 𝐻)
99	53	unssbd 4159	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⊆ 𝐶)
100	99	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐻 ⊆ 𝐶)
101	98, 100	sstrd 3959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ 𝐶)
102	70	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐶 = (Base‘𝐸))
103	101, 102	sseqtrd 3985	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ (Base‘𝐸))
104		eqidd 2731	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
105	72	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐺 ⊆ (Base‘𝐸))
106	104, 105	srabase 21090	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐺)))
107	103, 106	sseqtrd 3985	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘𝐺)))
108	61	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
109	89, 90	lspssv 20895	. . . . . . . 8 ⊢ ((((subringAlg ‘𝐸)‘𝐺) ∈ LMod ∧ 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘𝐺))) → ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏) ⊆ (Base‘((subringAlg ‘𝐸)‘𝐺)))
110	108, 107, 109	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏) ⊆ (Base‘((subringAlg ‘𝐸)‘𝐺)))
111		fldextrspunfld.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (𝐿 ↾_s 𝐹)
112	40	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐿 ∈ Field)
113	14	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐹 ∈ (SubDRing‘𝐼))
114	5	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐹 ∈ (SubDRing‘𝐽))
115	11	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐺 ∈ (SubDRing‘𝐿))
116	1	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐻 ∈ (SubDRing‘𝐿))
117		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
118		fldsdrgfld 20713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ Field ∧ 𝐻 ∈ (SubDRing‘𝐿)) → (𝐿 ↾_s 𝐻) ∈ Field)
119	40, 1, 118	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐿 ↾_s 𝐻) ∈ Field)
120	2, 119	eqeltrid 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽 ∈ Field)
121		ressabs 17224	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐻 ∈ (SubDRing‘𝐿) ∧ 𝐹 ⊆ 𝐻) → ((𝐿 ↾_s 𝐻) ↾_s 𝐹) = (𝐿 ↾_s 𝐹))
122	1, 30, 121	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐿 ↾_s 𝐻) ↾_s 𝐹) = (𝐿 ↾_s 𝐹))
123	2	oveq1i 7399	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ↾_s 𝐹) = ((𝐿 ↾_s 𝐻) ↾_s 𝐹)
124	122, 123, 111	3eqtr4g 2790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) = 𝐾)
125		fldsdrgfld 20713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐽)) → (𝐽 ↾_s 𝐹) ∈ Field)
126	120, 5, 125	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) ∈ Field)
127	124, 126	eqeltrrd 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Field)
128	30, 27	sstrd 3959	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
129	111, 25	ressbas2 17214	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ⊆ (Base‘𝐿) → 𝐹 = (Base‘𝐾))
130	128, 129	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
131	130	oveq2d 7405	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) = (𝐽 ↾_s (Base‘𝐾)))
132	124, 131	eqtr3d 2767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 = (𝐽 ↾_s (Base‘𝐾)))
133	130, 33	eqeltrrd 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐽))
134		brfldext 33647	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Field ∧ 𝐾 ∈ Field) → (𝐽/_FldExt𝐾 ↔ (𝐾 = (𝐽 ↾_s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐽))))
135	134	biimpar 477	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐽 ↾_s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐽))) → 𝐽/_FldExt𝐾)
136	120, 127, 132, 133, 135	syl22anc 838	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽/_FldExt𝐾)
137	136	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐽/_FldExt𝐾)
138		extdgval 33655	. . . . . . . . . . . . . . . . 17 ⊢ (𝐽/_FldExt𝐾 → (𝐽[:]𝐾) = (dim‘((subringAlg ‘𝐽)‘(Base‘𝐾))))
139	137, 138	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (𝐽[:]𝐾) = (dim‘((subringAlg ‘𝐽)‘(Base‘𝐾))))
140	130	fveq2d 6864	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘(Base‘𝐾)))
141	140	fveq2d 6864	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐽)‘𝐹)) = (dim‘((subringAlg ‘𝐽)‘(Base‘𝐾))))
142	141	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐽)‘𝐹)) = (dim‘((subringAlg ‘𝐽)‘(Base‘𝐾))))
143	37	dimval 33602	. . . . . . . . . . . . . . . . 17 ⊢ ((((subringAlg ‘𝐽)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐽)‘𝐹)) = (♯‘𝑏))
144	36, 143	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐽)‘𝐹)) = (♯‘𝑏))
145	139, 142, 144	3eqtr2d 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (𝐽[:]𝐾) = (♯‘𝑏))
146		fldextrspunfld.7	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
147	146	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (𝐽[:]𝐾) ∈ ℕ₀)
148	145, 147	eqeltrrd 2830	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (♯‘𝑏) ∈ ℕ₀)
149		hashclb 14329	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → (𝑏 ∈ Fin ↔ (♯‘𝑏) ∈ ℕ₀))
150	149	biimpar 477	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) ∧ (♯‘𝑏) ∈ ℕ₀) → 𝑏 ∈ Fin)
151	117, 148, 150	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ∈ Fin)
152	111, 17, 2, 112, 113, 114, 115, 116, 48, 50, 55, 117, 151	fldextrspunlsp 33675	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏))
153	152	eqimssd 4005	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 ⊆ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏))
154	25, 55, 68, 54, 40	resssra 33589	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = (((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))
155	154	fveq2d 6864	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (LSpan‘((subringAlg ‘𝐸)‘𝐺)) = (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)))
156	155	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (LSpan‘((subringAlg ‘𝐸)‘𝐺)) = (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)))
157	156	fveq1d 6862	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏) = ((LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))‘𝑏))
158	115, 12	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐺 ∈ (SubRing‘𝐿))
159		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺)
160	159	sralmod 21100	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (SubRing‘𝐿) → ((subringAlg ‘𝐿)‘𝐺) ∈ LMod)
161	158, 160	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((subringAlg ‘𝐿)‘𝐺) ∈ LMod)
162	117, 92	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
163	96	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
164	162, 163	sseqtrrd 3986	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ 𝐻)
165	116, 26	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐻 ⊆ (Base‘𝐿))
166	164, 165	sstrd 3959	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ (Base‘𝐿))
167		eqidd 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺))
168	167, 46	srabase 21090	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐺)))
169	168	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐺)))
170	166, 169	sseqtrd 3985	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐿)‘𝐺)))
171		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (Base‘((subringAlg ‘𝐿)‘𝐺)) = (Base‘((subringAlg ‘𝐿)‘𝐺))
172		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)) = (LSubSp‘((subringAlg ‘𝐿)‘𝐺))
173		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘((subringAlg ‘𝐿)‘𝐺)) = (LSpan‘((subringAlg ‘𝐿)‘𝐺))
174	171, 172, 173	lspcl 20888	. . . . . . . . . . . . . . 15 ⊢ ((((subringAlg ‘𝐿)‘𝐺) ∈ LMod ∧ 𝑏 ⊆ (Base‘((subringAlg ‘𝐿)‘𝐺))) → ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏) ∈ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)))
175	161, 170, 174	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏) ∈ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)))
176	152, 175	eqeltrd 2829	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 ∈ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)))
177	99	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐻 ⊆ 𝐶)
178	164, 177	sstrd 3959	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ 𝐶)
179		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶) = (((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)
180		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)) = (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))
181	179, 173, 180, 172	lsslsp 20927	. . . . . . . . . . . . 13 ⊢ ((((subringAlg ‘𝐿)‘𝐺) ∈ LMod ∧ 𝐶 ∈ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)) ∧ 𝑏 ⊆ 𝐶) → ((LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))‘𝑏) = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏))
182	161, 176, 178, 181	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))‘𝑏) = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏))
183	157, 182	eqtr2d 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏) = ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
184	153, 183	sseqtrd 3985	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 ⊆ ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
185	184	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐶 ⊆ ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
186	102, 185	eqsstrrd 3984	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (Base‘𝐸) ⊆ ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
187	106, 186	eqsstrrd 3984	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (Base‘((subringAlg ‘𝐸)‘𝐺)) ⊆ ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
188	110, 187	eqssd 3966	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏) = (Base‘((subringAlg ‘𝐸)‘𝐺)))
189	89, 81, 90, 85, 86, 107, 188	lbslelsp 33599	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (♯‘𝑐) ≤ (♯‘𝑏))
190	88, 189	eqbrtrd 5131	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (♯‘𝑏))
191	84, 190	n0limd 32407	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (♯‘𝑏))
192	191, 145	breqtrrd 5137	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))
193	39, 192	n0limd 32407	1 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∪ cun 3914 ⊆ wss 3916 ∅c0 4298 class class class wbr 5109 ‘cfv 6513 (class class class)co 7389 Fincfn 8920 ≤ cle 11215 ℕ₀cn0 12448 ♯chash 14301 Basecbs 17185 ↾_s cress 17206 Scalarcsca 17229 SubRingcsubrg 20484 RingSpancrgspn 20525 IDomncidom 20608 DivRingcdr 20644 Fieldcfield 20645 SubDRingcsdrg 20701 LModclmod 20772 LSubSpclss 20843 LSpanclspn 20883 LBasisclbs 20987 LVecclvec 21015 subringAlg csra 21084 dimcldim 33600 /_FldExtcfldext 33640 [:]cextdg 33642

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-reg 9551 ax-inf2 9600 ax-ac2 10422 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-rpss 7701 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-oadd 8440 df-er 8673 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-sup 9399 df-inf 9400 df-oi 9469 df-r1 9723 df-rank 9724 df-dju 9860 df-card 9898 df-acn 9901 df-ac 10075 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-rp 12958 df-fz 13475 df-fzo 13622 df-seq 13973 df-exp 14033 df-hash 14302 df-word 14485 df-lsw 14534 df-concat 14542 df-s1 14567 df-substr 14612 df-pfx 14642 df-s2 14820 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-clim 15460 df-sum 15659 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ocomp 17247 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17410 df-gsum 17411 df-prds 17416 df-pws 17418 df-mre 17553 df-mrc 17554 df-mri 17555 df-acs 17556 df-proset 18261 df-drs 18262 df-poset 18280 df-ipo 18493 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18716 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-mulg 19006 df-subg 19061 df-ghm 19151 df-cntz 19255 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-ring 20150 df-cring 20151 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rgspn 20526 df-rlreg 20609 df-domn 20610 df-idom 20611 df-drng 20646 df-field 20647 df-sdrg 20702 df-lmod 20774 df-lss 20844 df-lsp 20884 df-lmhm 20935 df-lbs 20988 df-lvec 21016 df-sra 21086 df-rgmod 21087 df-cnfld 21271 df-zring 21363 df-dsmm 21647 df-frlm 21662 df-uvc 21698 df-ind 32780 df-dim 33601 df-fldext 33643 df-extdg 33644

This theorem is referenced by: fldextrspunfld 33677 fldextrspundgle 33679

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