fldextrspunlem1 - Mathbox for Thierry Arnoux

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Theorem fldextrspunlem1 34355

Description: Lemma for fldextrspunfld 34356. 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 21054	. . . . 5 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐽 ∈ DivRing)
4	1, 3	syl 18	. . . 4 ⊢ (𝜑 → 𝐽 ∈ DivRing)
5		fldextrspunfld.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
6		eqid 2760	. . . . . 6 ⊢ (𝐽 ↾_s 𝐹) = (𝐽 ↾_s 𝐹)
7	6	sdrgdrng 21054	. . . . 5 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → (𝐽 ↾_s 𝐹) ∈ DivRing)
8	5, 7	syl 18	. . . 4 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) ∈ DivRing)
9		sdrgsubrg 21055	. . . . . 6 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ∈ (SubRing‘𝐿))
10	1, 9	syl 18	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (SubRing‘𝐿))
11		fldextrspunfld.5	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
12		sdrgsubrg 21055	. . . . . . . 8 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
13	11, 12	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
14		fldextrspunfld.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
15		sdrgsubrg 21055	. . . . . . . 8 ⊢ (𝐹 ∈ (SubDRing‘𝐼) → 𝐹 ∈ (SubRing‘𝐼))
16	14, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐼))
17		fldextrspunfld.i	. . . . . . . . 9 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
18	17	subsubrg 20857	. . . . . . . 8 ⊢ (𝐺 ∈ (SubRing‘𝐿) → (𝐹 ∈ (SubRing‘𝐼) ↔ (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐺)))
19	18	biimpa 482	. . . . . . 7 ⊢ ((𝐺 ∈ (SubRing‘𝐿) ∧ 𝐹 ∈ (SubRing‘𝐼)) → (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐺))
20	13, 16, 19	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐺))
21	20	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿))
22		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘𝐽)
23	22	sdrgss 21057	. . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
24	5, 23	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
25		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝐿) = (Base‘𝐿)
26	25	sdrgss 21057	. . . . . . . 8 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
27	1, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
28	2, 25	ressbas2 17423	. . . . . . 7 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
29	27, 28	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
30	24, 29	sseqtrrd 3946	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ 𝐻)
31	2	subsubrg 20857	. . . . . 6 ⊢ (𝐻 ∈ (SubRing‘𝐿) → (𝐹 ∈ (SubRing‘𝐽) ↔ (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐻)))
32	31	biimpar 483	. . . . 5 ⊢ ((𝐻 ∈ (SubRing‘𝐿) ∧ (𝐹 ∈ (SubRing‘𝐿) ∧ 𝐹 ⊆ 𝐻)) → 𝐹 ∈ (SubRing‘𝐽))
33	10, 21, 30, 32	syl12anc 850	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐽))
34		eqid 2760	. . . . 5 ⊢ ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹)
35	34, 6	sralvec 34265	. . . 4 ⊢ ((𝐽 ∈ DivRing ∧ (𝐽 ↾_s 𝐹) ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐽)) → ((subringAlg ‘𝐽)‘𝐹) ∈ LVec)
36	4, 8, 33, 35	syl3anc 1398	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) ∈ LVec)
37		eqid 2760	. . . 4 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
38	37	lbsex 21450	. . 3 ⊢ (((subringAlg ‘𝐽)‘𝐹) ∈ LVec → (LBasis‘((subringAlg ‘𝐽)‘𝐹)) ≠ ∅)
39	36, 38	syl 18	. 2 ⊢ (𝜑 → (LBasis‘((subringAlg ‘𝐽)‘𝐹)) ≠ ∅)
40		fldextrspunfld.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ Field)
41		fldidom 21036	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ Field → 𝐿 ∈ IDomn)
42	40, 41	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ IDomn)
43	42	idomringd 20986	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ Ring)
44		eqidd 2761	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
45	25	sdrgss 21057	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
46	11, 45	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
47	46, 27	unssd 4189	. . . . . . . . . 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 20872	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝐿))
53	43, 44, 47, 49, 51	rgspnssid 20873	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
54	53	unssad 4190	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
55		fldextrspunfld.e	. . . . . . . . . . 11 ⊢ 𝐸 = (𝐿 ↾_s 𝐶)
56	55	subsubrg 20857	. . . . . . . . . 10 ⊢ (𝐶 ∈ (SubRing‘𝐿) → (𝐺 ∈ (SubRing‘𝐸) ↔ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)))
57	56	biimpar 483	. . . . . . . . 9 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)) → 𝐺 ∈ (SubRing‘𝐸))
58	52, 13, 54, 57	syl12anc 850	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐸))
59		eqid 2760	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺)
60	59	sralmod 21469	. . . . . . . 8 ⊢ (𝐺 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
61	58, 60	syl 18	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
62		ressabs 17433	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶) → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
63	52, 54, 62	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
64	55	oveq1i 7432	. . . . . . . . . 10 ⊢ (𝐸 ↾_s 𝐺) = ((𝐿 ↾_s 𝐶) ↾_s 𝐺)
65	63, 64, 17	3eqtr4g 2820	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = 𝐼)
66		eqidd 2761	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
67	25	subrgss 20831	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ (SubRing‘𝐿) → 𝐶 ⊆ (Base‘𝐿))
68	52, 67	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐿))
69	55, 25	ressbas2 17423	. . . . . . . . . . . . 13 ⊢ (𝐶 ⊆ (Base‘𝐿) → 𝐶 = (Base‘𝐸))
70	68, 69	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 = (Base‘𝐸))
71	53, 70	sseqtrd 3945	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ (Base‘𝐸))
72	71	unssad 4190	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐸))
73	66, 72	srasca 21462	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
74	65, 73	eqtr3d 2797	. . . . . . . 8 ⊢ (𝜑 → 𝐼 = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
75	17	sdrgdrng 21054	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐼 ∈ DivRing)
76	11, 75	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ DivRing)
77	74, 76	eqeltrrd 2861	. . . . . . 7 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing)
78		eqid 2760	. . . . . . . 8 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐺)) = (Scalar‘((subringAlg ‘𝐸)‘𝐺))
79	78	islvec 21386	. . . . . . 7 ⊢ (((subringAlg ‘𝐸)‘𝐺) ∈ LVec ↔ (((subringAlg ‘𝐸)‘𝐺) ∈ LMod ∧ (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing))
80	61, 77, 79	sylanbrc 595	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LVec)
81		eqid 2760	. . . . . . 7 ⊢ (LBasis‘((subringAlg ‘𝐸)‘𝐺)) = (LBasis‘((subringAlg ‘𝐸)‘𝐺))
82	81	lbsex 21450	. . . . . 6 ⊢ (((subringAlg ‘𝐸)‘𝐺) ∈ LVec → (LBasis‘((subringAlg ‘𝐸)‘𝐺)) ≠ ∅)
83	80, 82	syl 18	. . . . 5 ⊢ (𝜑 → (LBasis‘((subringAlg ‘𝐸)‘𝐺)) ≠ ∅)
84	83	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (LBasis‘((subringAlg ‘𝐸)‘𝐺)) ≠ ∅)
85	80	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → ((subringAlg ‘𝐸)‘𝐺) ∈ LVec)
86		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺)))
87	81	dimval 34281	. . . . . 6 ⊢ ((((subringAlg ‘𝐸)‘𝐺) ∈ LVec ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) = (♯‘𝑐))
88	85, 86, 87	syl2anc 596	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) = (♯‘𝑐))
89		eqid 2760	. . . . . 6 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐺)) = (Base‘((subringAlg ‘𝐸)‘𝐺))
90		eqid 2760	. . . . . 6 ⊢ (LSpan‘((subringAlg ‘𝐸)‘𝐺)) = (LSpan‘((subringAlg ‘𝐸)‘𝐺))
91		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
92	91, 37	lbsss 21359	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
93	92	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
94		eqidd 2761	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
95	94, 24	srabase 21459	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
96	29, 95	eqtrd 2795	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
97	96	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
98	93, 97	sseqtrrd 3946	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ 𝐻)
99	53	unssbd 4191	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⊆ 𝐶)
100	99	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐻 ⊆ 𝐶)
101	98, 100	sstrd 3919	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ 𝐶)
102	70	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐶 = (Base‘𝐸))
103	101, 102	sseqtrd 3945	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ (Base‘𝐸))
104		eqidd 2761	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
105	72	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐺 ⊆ (Base‘𝐸))
106	104, 105	srabase 21459	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐺)))
107	103, 106	sseqtrd 3945	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘𝐺)))
108	61	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
109	89, 90	lspssv 21265	. . . . . . . 8 ⊢ ((((subringAlg ‘𝐸)‘𝐺) ∈ LMod ∧ 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘𝐺))) → ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏) ⊆ (Base‘((subringAlg ‘𝐸)‘𝐺)))
110	108, 107, 109	syl2anc 596	. . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐿 ∈ Field)
113	14	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐹 ∈ (SubDRing‘𝐼))
114	5	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐹 ∈ (SubDRing‘𝐽))
115	11	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐺 ∈ (SubDRing‘𝐿))
116	1	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐻 ∈ (SubDRing‘𝐿))
117		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
118		fldsdrgfld 21062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐿 ∈ Field ∧ 𝐻 ∈ (SubDRing‘𝐿)) → (𝐿 ↾_s 𝐻) ∈ Field)
119	40, 1, 118	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐿 ↾_s 𝐻) ∈ Field)
120	2, 119	eqeltrid 2864	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐽 ∈ Field)
121		ressabs 17433	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐻 ∈ (SubDRing‘𝐿) ∧ 𝐹 ⊆ 𝐻) → ((𝐿 ↾_s 𝐻) ↾_s 𝐹) = (𝐿 ↾_s 𝐹))
122	1, 30, 121	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐿 ↾_s 𝐻) ↾_s 𝐹) = (𝐿 ↾_s 𝐹))
123	2	oveq1i 7432	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ↾_s 𝐹) = ((𝐿 ↾_s 𝐻) ↾_s 𝐹)
124	122, 123, 111	3eqtr4g 2820	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) = 𝐾)
125		fldsdrgfld 21062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐽)) → (𝐽 ↾_s 𝐹) ∈ Field)
126	120, 5, 125	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) ∈ Field)
127	124, 126	eqeltrrd 2861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 ∈ Field)
128	30, 27	sstrd 3919	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
129	111, 25	ressbas2 17423	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ⊆ (Base‘𝐿) → 𝐹 = (Base‘𝐾))
130	128, 129	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
131	130	oveq2d 7438	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 ↾_s 𝐹) = (𝐽 ↾_s (Base‘𝐾)))
132	124, 131	eqtr3d 2797	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾 = (𝐽 ↾_s (Base‘𝐾)))
133	130, 33	eqeltrrd 2861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐽))
134		brfldext 34325	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Field ∧ 𝐾 ∈ Field) → (𝐽/_FldExt𝐾 ↔ (𝐾 = (𝐽 ↾_s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐽))))
135	134	biimpar 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐽 ↾_s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐽))) → 𝐽/_FldExt𝐾)
136	120, 127, 132, 133, 135	syl22anc 852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽/_FldExt𝐾)
137	136	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐽/_FldExt𝐾)
138		extdgval 34333	. . . . . . . . . . . . . . . . 17 ⊢ (𝐽/_FldExt𝐾 → (𝐽[:]𝐾) = (dim‘((subringAlg ‘𝐽)‘(Base‘𝐾))))
139	137, 138	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (𝐽[:]𝐾) = (dim‘((subringAlg ‘𝐽)‘(Base‘𝐾))))
140	130	fveq2d 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘(Base‘𝐾)))
141	140	fveq2d 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐽)‘𝐹)) = (dim‘((subringAlg ‘𝐽)‘(Base‘𝐾))))
142	141	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐽)‘𝐹)) = (dim‘((subringAlg ‘𝐽)‘(Base‘𝐾))))
143	37	dimval 34281	. . . . . . . . . . . . . . . . 17 ⊢ ((((subringAlg ‘𝐽)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐽)‘𝐹)) = (♯‘𝑏))
144	36, 143	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐽)‘𝐹)) = (♯‘𝑏))
145	139, 142, 144	3eqtr2d 2801	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (𝐽[:]𝐾) = (♯‘𝑏))
146		fldextrspunfld.7	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
147	146	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (𝐽[:]𝐾) ∈ ℕ₀)
148	145, 147	eqeltrrd 2861	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (♯‘𝑏) ∈ ℕ₀)
149		hashclb 14507	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → (𝑏 ∈ Fin ↔ (♯‘𝑏) ∈ ℕ₀))
150	149	biimpar 483	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) ∧ (♯‘𝑏) ∈ ℕ₀) → 𝑏 ∈ Fin)
151	117, 148, 150	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ∈ Fin)
152	111, 17, 2, 112, 113, 114, 115, 116, 48, 50, 55, 117, 151	fldextrspunlsp 34354	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏))
153	152	eqimssd 3965	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 ⊆ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏))
154	25, 55, 68, 54, 40	resssra 34267	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = (((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))
155	154	fveq2d 6892	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (LSpan‘((subringAlg ‘𝐸)‘𝐺)) = (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)))
156	155	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (LSpan‘((subringAlg ‘𝐸)‘𝐺)) = (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)))
157	156	fveq1d 6890	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏) = ((LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))‘𝑏))
158	115, 12	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐺 ∈ (SubRing‘𝐿))
159		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺)
160	159	sralmod 21469	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (SubRing‘𝐿) → ((subringAlg ‘𝐿)‘𝐺) ∈ LMod)
161	158, 160	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((subringAlg ‘𝐿)‘𝐺) ∈ LMod)
162	117, 92	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
163	96	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
164	162, 163	sseqtrrd 3946	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ 𝐻)
165	116, 26	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐻 ⊆ (Base‘𝐿))
166	164, 165	sstrd 3919	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ (Base‘𝐿))
167		eqidd 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺))
168	167, 46	srabase 21459	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐺)))
169	168	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐺)))
170	166, 169	sseqtrd 3945	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐿)‘𝐺)))
171		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (Base‘((subringAlg ‘𝐿)‘𝐺)) = (Base‘((subringAlg ‘𝐿)‘𝐺))
172		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)) = (LSubSp‘((subringAlg ‘𝐿)‘𝐺))
173		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘((subringAlg ‘𝐿)‘𝐺)) = (LSpan‘((subringAlg ‘𝐿)‘𝐺))
174	171, 172, 173	lspcl 21258	. . . . . . . . . . . . . . 15 ⊢ ((((subringAlg ‘𝐿)‘𝐺) ∈ LMod ∧ 𝑏 ⊆ (Base‘((subringAlg ‘𝐿)‘𝐺))) → ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏) ∈ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)))
175	161, 170, 174	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏) ∈ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)))
176	152, 175	eqeltrd 2860	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 ∈ (LSubSp‘((subringAlg ‘𝐿)‘𝐺)))
177	99	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐻 ⊆ 𝐶)
178	164, 177	sstrd 3919	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝑏 ⊆ 𝐶)
179		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶) = (((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)
180		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶)) = (LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))
181	179, 173, 180, 172	lsslsp 21297	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((LSpan‘(((subringAlg ‘𝐿)‘𝐺) ↾_s 𝐶))‘𝑏) = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏))
183	157, 182	eqtr2d 2796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝑏) = ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
184	153, 183	sseqtrd 3945	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → 𝐶 ⊆ ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
185	184	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → 𝐶 ⊆ ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
186	102, 185	eqsstrrd 3944	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (Base‘𝐸) ⊆ ((LSpan‘((subringAlg ‘𝐸)‘𝐺))‘𝑏))
187	106, 186	eqsstrrd 3944	. . . . . . 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 3926	. . . . . 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 34278	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (♯‘𝑐) ≤ (♯‘𝑏))
190	88, 189	eqbrtrd 5126	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑐 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐺))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (♯‘𝑏))
191	84, 190	n0limd 4300	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (♯‘𝑏))
192	191, 145	breqtrrd 5132	. 2 ⊢ ((𝜑 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))
193	39, 192	n0limd 4300	1 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ⊆ wss 3896 ∪ cun 4148 ∅c0 4277 class class class wbr 5102 ‘cfv 6541 (class class class)co 7422 Fincfn 8978 ≤ cle 11349 ℕ₀cn0 12611 ♯chash 14479 Basecbs 17394 ↾_s cress 17415 Scalarcsca 17438 SubRingcsubrg 20828 RingSpancrgspn 20869 IDomncidom 20952 DivRingcdr 20987 Fieldcfield 20988 SubDRingcsdrg 21050 LModclmod 21142 LSubSpclss 21213 LSpanclspn 21253 LBasisclbs 21356 LVecclvec 21384 subringAlg csra 21453 dimcldim 34279 /_FldExtcfldext 34318 [:]cextdg 34320

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-inf2 9647 ax-ac2 10546 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-pre-sup 11283 ax-addf 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-of 7695 df-rpss 7741 df-om 7880 df-1st 8003 df-2nd 8004 df-supp 8182 df-tpos 8248 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-2o 8482 df-oadd 8485 df-er 8722 df-map 8854 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9359 df-sup 9439 df-inf 9440 df-oi 9509 df-dju 9987 df-card 10025 df-acn 10028 df-ac 10200 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-ind 12326 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-xnn0 12685 df-z 12699 df-dec 12820 df-uz 12971 df-rp 13126 df-fz 13645 df-fzo 13794 df-seq 14150 df-exp 14210 df-hash 14480 df-word 14664 df-lsw 14713 df-concat 14721 df-s1 14748 df-substr 14795 df-pfx 14827 df-s2 15005 df-cj 15273 df-re 15274 df-im 15275 df-sqrt 15409 df-abs 15410 df-clim 15662 df-sum 15861 df-struct 17332 df-sets 17349 df-slot 17367 df-ndx 17379 df-base 17395 df-ress 17416 df-plusg 17448 df-mulr 17449 df-starv 17450 df-sca 17451 df-vsca 17452 df-ip 17453 df-tset 17454 df-ple 17455 df-ocomp 17456 df-ds 17457 df-unif 17458 df-hom 17459 df-cco 17460 df-0g 17619 df-gsum 17620 df-prds 17625 df-pws 17627 df-mre 17763 df-mrc 17764 df-mri 17765 df-acs 17766 df-proset 18475 df-drs 18476 df-poset 18494 df-ipo 18709 df-mgm 18823 df-sgrp 18915 df-mnd 18931 df-mhm 18985 df-submnd 18986 df-grp 19154 df-minusg 19155 df-sbg 19156 df-mulg 19285 df-subg 19340 df-ghm 19435 df-cntz 19538 df-cmn 20003 df-abl 20004 df-mgp 20368 df-rng 20382 df-ur 20415 df-ring 20468 df-cring 20469 df-oppr 20574 df-dvdsr 20594 df-unit 20595 df-invr 20625 df-nzr 20770 df-subrng 20805 df-subrg 20829 df-rgspn 20870 df-rlreg 20953 df-domn 20954 df-idom 20955 df-drng 20989 df-field 20990 df-sdrg 21051 df-lmod 21144 df-lss 21214 df-lsp 21254 df-lmhm 21304 df-lbs 21357 df-lvec 21385 df-sra 21455 df-rgmod 21456 df-cnfld 21686 df-zring 21760 df-dsmm 22045 df-frlm 22060 df-uvc 22096 df-dim 34280 df-fldext 34321 df-extdg 34322

This theorem is used by: fldextrspunfld 34356 fldextrspundgle 34358

W3C validator