fldextrspunlem1 - Mathbox for Thierry Arnoux

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Theorem fldextrspunlem1 33832

Description: Lemma for fldextrspunfld 33833. 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 20723	. . . . 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 2736	. . . . . 6 ⊢ (𝐽 ↾_s 𝐹) = (𝐽 ↾_s 𝐹)
7	6	sdrgdrng 20723	. . . . 5 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → (𝐽 ↾_s 𝐹) ∈ DivRing)
8	5, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) ∈ DivRing)
9		sdrgsubrg 20724	. . . . . 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 20724	. . . . . . . 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 20724	. . . . . . . 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 20531	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘𝐽)
23	22	sdrgss 20726	. . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
24	5, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
25		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐿) = (Base‘𝐿)
26	25	sdrgss 20726	. . . . . . . 8 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
27	1, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
28	2, 25	ressbas2 17165	. . . . . . 7 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
29	27, 28	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
30	24, 29	sseqtrrd 3971	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ 𝐻)
31	2	subsubrg 20531	. . . . . 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 2736	. . . . 5 ⊢ ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹)
35	34, 6	sralvec 33741	. . . 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 2736	. . . 4 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
38	37	lbsex 21120	. . 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 20704	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Field → 𝐿 ∈ IDomn)
42	40, 41	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ IDomn)
43	42	idomringd 20661	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ Ring)
44		eqidd 2737	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
45	25	sdrgss 20726	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
46	11, 45	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
47	46, 27	unssd 4144	. . . . . . . . . 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 20546	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝐿))
53	43, 44, 47, 49, 51	rgspnssid 20547	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
54	53	unssad 4145	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
55		fldextrspunfld.e	. . . . . . . . . . 11 ⊢ 𝐸 = (𝐿 ↾_s 𝐶)
56	55	subsubrg 20531	. . . . . . . . . 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 2736	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺)
60	59	sralmod 21139	. . . . . . . 8 ⊢ (𝐺 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
61	58, 60	syl 17	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
62		ressabs 17175	. . . . . . . . . . 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 7368	. . . . . . . . . 10 ⊢ (𝐸 ↾_s 𝐺) = ((𝐿 ↾_s 𝐶) ↾_s 𝐺)
65	63, 64, 17	3eqtr4g 2796	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = 𝐼)
66		eqidd 2737	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
67	25	subrgss 20505	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (SubRing‘𝐿) → 𝐶 ⊆ (Base‘𝐿))
68	52, 67	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐿))
69	55, 25	ressbas2 17165	. . . . . . . . . . . . 13 ⊢ (𝐶 ⊆ (Base‘𝐿) → 𝐶 = (Base‘𝐸))
70	68, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 = (Base‘𝐸))
71	53, 70	sseqtrd 3970	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ (Base‘𝐸))
72	71	unssad 4145	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐸))
73	66, 72	srasca 21132	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
74	65, 73	eqtr3d 2773	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
75	17	sdrgdrng 20723	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐼 ∈ DivRing)
76	11, 75	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ DivRing)
77	74, 76	eqeltrrd 2837	. . . . . . 7 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing)
78		eqid 2736	. . . . . . . 8 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐺)) = (Scalar‘((subringAlg ‘𝐸)‘𝐺))
79	78	islvec 21056	. . . . . . 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 2736	. . . . . . 7 ⊢ (LBasis‘((subringAlg ‘𝐸)‘𝐺)) = (LBasis‘((subringAlg ‘𝐸)‘𝐺))
82	81	lbsex 21120	. . . . . 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 33757	. . . . . 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 2736	. . . . . 6 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐺)) = (Base‘((subringAlg ‘𝐸)‘𝐺))
90		eqid 2736	. . . . . 6 ⊢ (LSpan‘((subringAlg ‘𝐸)‘𝐺)) = (LSpan‘((subringAlg ‘𝐸)‘𝐺))
91		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
92	91, 37	lbsss 21029	. . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
95	94, 24	srabase 21129	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
96	29, 95	eqtrd 2771	. . . . . . . . . . 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 3971	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ 𝐻)
99	53	unssbd 4146	. . . . . . . . . 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 3944	. . . . . . . 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 3970	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ (Base‘𝐸))
104		eqidd 2737	. . . . . . . 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 21129	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐺)))
107	103, 106	sseqtrd 3970	. . . . . 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 20934	. . . . . . . 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 20731	. . . . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽 ∈ Field)
121		ressabs 17175	. . . . . . . . . . . . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ↾_s 𝐹) = ((𝐿 ↾_s 𝐻) ↾_s 𝐹)
124	122, 123, 111	3eqtr4g 2796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) = 𝐾)
125		fldsdrgfld 20731	. . . . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Field)
128	30, 27	sstrd 3944	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
129	111, 25	ressbas2 17165	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ⊆ (Base‘𝐿) → 𝐹 = (Base‘𝐾))
130	128, 129	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
131	130	oveq2d 7374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) = (𝐽 ↾_s (Base‘𝐾)))
132	124, 131	eqtr3d 2773	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 = (𝐽 ↾_s (Base‘𝐾)))
133	130, 33	eqeltrrd 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐽))
134		brfldext 33802	. . . . . . . . . . . . . . . . . . . 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 33810	. . . . . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘(Base‘𝐾)))
141	140	fveq2d 6838	. . . . . . . . . . . . . . . . 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 33757	. . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (♯‘𝑏) ∈ ℕ₀)
149		hashclb 14281	. . . . . . . . . . . . . . 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 33831	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏))
153	152	eqimssd 3990	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 ⊆ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏))
154	25, 55, 68, 54, 40	resssra 33743	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = (((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))
155	154	fveq2d 6838	. . . . . . . . . . . . . 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 6836	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺)
160	159	sralmod 21139	. . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ (Base‘𝐿))
167		eqidd 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺))
168	167, 46	srabase 21129	. . . . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐿)‘𝐺)))
171		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (Base‘((subringAlg ‘𝐿)‘𝐺)) = (Base‘((subringAlg ‘𝐿)‘𝐺))
172		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)) = (LSubSp‘((subringAlg ‘𝐿)‘𝐺))
173		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘((subringAlg ‘𝐿)‘𝐺)) = (LSpan‘((subringAlg ‘𝐿)‘𝐺))
174	171, 172, 173	lspcl 20927	. . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ 𝐶)
179		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶) = (((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)
180		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)) = (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))
181	179, 173, 180, 172	lsslsp 20966	. . . . . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏) = ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
184	153, 183	sseqtrd 3970	. . . . . . . . . 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 3969	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (Base‘𝐸) ⊆ ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
187	106, 186	eqsstrrd 3969	. . . . . . 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 3951	. . . . . 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 33754	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (♯‘𝑐) ≤ (♯‘𝑏))
190	88, 189	eqbrtrd 5120	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (♯‘𝑏))
191	84, 190	n0limd 32546	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (♯‘𝑏))
192	191, 145	breqtrrd 5126	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))
193	39, 192	n0limd 32546	1 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∪ cun 3899 ⊆ wss 3901 ∅c0 4285 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 Fincfn 8883 ≤ cle 11167 ℕ₀cn0 12401 ♯chash 14253 Basecbs 17136 ↾_s cress 17157 Scalarcsca 17180 SubRingcsubrg 20502 RingSpancrgspn 20543 IDomncidom 20626 DivRingcdr 20662 Fieldcfield 20663 SubDRingcsdrg 20719 LModclmod 20811 LSubSpclss 20882 LSpanclspn 20922 LBasisclbs 21026 LVecclvec 21054 subringAlg csra 21123 dimcldim 33755 /_FldExtcfldext 33795 [:]cextdg 33797

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-reg 9497 ax-inf2 9550 ax-ac2 10373 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-rpss 7668 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-inf 9346 df-oi 9415 df-r1 9676 df-rank 9677 df-dju 9813 df-card 9851 df-acn 9854 df-ac 10026 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-rp 12906 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-word 14437 df-lsw 14486 df-concat 14494 df-s1 14520 df-substr 14565 df-pfx 14595 df-s2 14771 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ocomp 17198 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-0g 17361 df-gsum 17362 df-prds 17367 df-pws 17369 df-mre 17505 df-mrc 17506 df-mri 17507 df-acs 17508 df-proset 18217 df-drs 18218 df-poset 18236 df-ipo 18451 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-ghm 19142 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-nzr 20446 df-subrng 20479 df-subrg 20503 df-rgspn 20544 df-rlreg 20627 df-domn 20628 df-idom 20629 df-drng 20664 df-field 20665 df-sdrg 20720 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lmhm 20974 df-lbs 21027 df-lvec 21055 df-sra 21125 df-rgmod 21126 df-cnfld 21310 df-zring 21402 df-dsmm 21687 df-frlm 21702 df-uvc 21738 df-ind 32930 df-dim 33756 df-fldext 33798 df-extdg 33799

This theorem is referenced by: fldextrspunfld 33833 fldextrspundgle 33835

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