Theorem sltmuls2 28567

Description: One surreal set less-than relationship for cuts of 𝐴 and 𝐵. (Contributed by Scott Fenton, 7-Mar-2025.)

Hypotheses

Ref	Expression
sltmuls2.1	⊢ (𝜑 → 𝐿 <<s 𝑅)
sltmuls2.2	⊢ (𝜑 → 𝑀 <<s 𝑆)
sltmuls2.3	⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
sltmuls2.4	⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))

Assertion

Ref	Expression
sltmuls2	⊢ (𝜑 → {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}))

Distinct variable groups:   𝐴,𝑐   𝐴,𝑑   𝑡,𝐴,𝑢   𝑣,𝐴,𝑤   𝐵,𝑐   𝐵,𝑑   𝑡,𝐵,𝑢   𝑣,𝐵,𝑤   𝐿,𝑐,𝑡,𝑢   𝑀,𝑑,𝑣,𝑤   𝑅,𝑑,𝑣,𝑤   𝑆,𝑐,𝑡,𝑢   𝜑,𝑐,𝑡,𝑢   𝜑,𝑑,𝑣,𝑤

Allowed substitution hints:   𝑅(𝑢, 𝑡, 𝑐)   𝑆(𝑤, 𝑣, 𝑑)   𝐿(𝑤, 𝑣, 𝑑)   𝑀(𝑢, 𝑡, 𝑐)

Proof of Theorem sltmuls2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5402	. . 3 ⊢ {(𝐴 ·s 𝐵)} ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → {(𝐴 ·s 𝐵)} ∈ V)
3		eqid 2763	. . . . 5 ⊢ (𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))) = (𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)))
4	3	rnmpo 7580	. . . 4 ⊢ ran (𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))) = {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))}
5		sltmuls2.1	. . . . . . 7 ⊢ (𝜑 → 𝐿 <<s 𝑅)
6		sltsex1 28182	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝐿 ∈ V)
7	5, 6	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ V)
8		sltmuls2.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 <<s 𝑆)
9		sltsex2 28183	. . . . . . 7 ⊢ (𝑀 <<s 𝑆 → 𝑆 ∈ V)
10	8, 9	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
11	3	mpoexg 8117	. . . . . 6 ⊢ ((𝐿 ∈ V ∧ 𝑆 ∈ V) → (𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))) ∈ V)
12	7, 10, 11	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))) ∈ V)
13		rnexg 7941	. . . . 5 ⊢ ((𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))) ∈ V → ran (𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))) ∈ V)
14	12, 13	syl 18	. . . 4 ⊢ (𝜑 → ran (𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))) ∈ V)
15	4, 14	eqeltrrid 2869	. . 3 ⊢ (𝜑 → {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∈ V)
16		eqid 2763	. . . . 5 ⊢ (𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))) = (𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)))
17	16	rnmpo 7580	. . . 4 ⊢ ran (𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))) = {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}
18		sltsex2 28183	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝑅 ∈ V)
19	5, 18	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
20		sltsex1 28182	. . . . . . 7 ⊢ (𝑀 <<s 𝑆 → 𝑀 ∈ V)
21	8, 20	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ V)
22	16	mpoexg 8117	. . . . . 6 ⊢ ((𝑅 ∈ V ∧ 𝑀 ∈ V) → (𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))) ∈ V)
23	19, 21, 22	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))) ∈ V)
24		rnexg 7941	. . . . 5 ⊢ ((𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))) ∈ V → ran (𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))) ∈ V)
25	23, 24	syl 18	. . . 4 ⊢ (𝜑 → ran (𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))) ∈ V)
26	17, 25	eqeltrrid 2869	. . 3 ⊢ (𝜑 → {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))} ∈ V)
27	15, 26	unexd 7795	. 2 ⊢ (𝜑 → ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) ∈ V)
28		sltmuls2.3	. . . . 5 ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
29	5	cutscld 28202	. . . . 5 ⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No)
30	28, 29	eqeltrd 2864	. . . 4 ⊢ (𝜑 → 𝐴 ∈ No)
31		sltmuls2.4	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))
32	8	cutscld 28202	. . . . 5 ⊢ (𝜑 → (𝑀 |s 𝑆) ∈ No)
33	31, 32	eqeltrd 2864	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No)
34	30, 33	mulscld 28554	. . 3 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No)
35	34	snssd 4747	. 2 ⊢ (𝜑 → {(𝐴 ·s 𝐵)} ⊆ No)
36		sltsss1 28184	. . . . . . . . . . . 12 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No)
37	5, 36	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ⊆ No)
38	37	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐿 ⊆ No)
39		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ 𝐿)
40	38, 39	sseldd 3914	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ No)
41	33	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ No)
42	40, 41	mulscld 28554	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝐵) ∈ No)
43	30	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐴 ∈ No)
44		sltsss2 28185	. . . . . . . . . . . 12 ⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆ No)
45	8, 44	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ No)
46	45	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ⊆ No)
47		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆)
48	46, 47	sseldd 3914	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ No)
49	43, 48	mulscld 28554	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝑢) ∈ No)
50	42, 49	addscld 28399	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No)
51	40, 48	mulscld 28554	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝑢) ∈ No)
52	50, 51	subscld 28482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) ∈ No)
53		eleq1 2852	. . . . . 6 ⊢ (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) → (𝑐 ∈ No ↔ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) ∈ No))
54	52, 53	syl5ibrcom 250	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) → 𝑐 ∈ No))
55	54	rexlimdvva 3223	. . . 4 ⊢ (𝜑 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) → 𝑐 ∈ No))
56	55	abssdv 3995	. . 3 ⊢ (𝜑 → {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ⊆ No)
57		sltsss2 28185	. . . . . . . . . . . 12 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No)
58	5, 57	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ⊆ No)
59	58	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑅 ⊆ No)
60		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ 𝑅)
61	59, 60	sseldd 3914	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ No)
62	33	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ No)
63	61, 62	mulscld 28554	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝐵) ∈ No)
64	30	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ No)
65		sltsss1 28184	. . . . . . . . . . . 12 ⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆ No)
66	8, 65	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ⊆ No)
67	66	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑀 ⊆ No)
68		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ 𝑀)
69	67, 68	sseldd 3914	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ No)
70	64, 69	mulscld 28554	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝑤) ∈ No)
71	63, 70	addscld 28399	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) ∈ No)
72	61, 69	mulscld 28554	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝑤) ∈ No)
73	71, 72	subscld 28482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) ∈ No)
74		eleq1 2852	. . . . . 6 ⊢ (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) → (𝑑 ∈ No ↔ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) ∈ No))
75	73, 74	syl5ibrcom 250	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) → 𝑑 ∈ No))
76	75	rexlimdvva 3223	. . . 4 ⊢ (𝜑 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) → 𝑑 ∈ No))
77	76	abssdv 3995	. . 3 ⊢ (𝜑 → {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))} ⊆ No)
78	56, 77	unssd 4193	. 2 ⊢ (𝜑 → ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) ⊆ No)
79		elun 4155	. . . . . 6 ⊢ (𝑦 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) ↔ (𝑦 ∈ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∨ 𝑦 ∈ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}))
80		vex 3458	. . . . . . . 8 ⊢ 𝑦 ∈ V
81		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑐 = 𝑦 → (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) ↔ 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))))
82	81	2rexbidv 3231	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))))
83	80, 82	elab 3636	. . . . . . 7 ⊢ (𝑦 ∈ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)))
84		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑑 = 𝑦 → (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) ↔ 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))))
85	84	2rexbidv 3231	. . . . . . . 8 ⊢ (𝑑 = 𝑦 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))))
86	80, 85	elab 3636	. . . . . . 7 ⊢ (𝑦 ∈ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))} ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)))
87	83, 86	orbi12i 928	. . . . . 6 ⊢ ((𝑦 ∈ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∨ 𝑦 ∈ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) ↔ (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) ∨ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))))
88	79, 87	bitri 278	. . . . 5 ⊢ (𝑦 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) ↔ (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) ∨ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))))
89		cutcuts 28200	. . . . . . . . . . . . . . 15 ⊢ (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
90	5, 89	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
91	90	simp2d 1161	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 <<s {(𝐿 |s 𝑅)})
92	91	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐿 <<s {(𝐿 |s 𝑅)})
93		ovex 7479	. . . . . . . . . . . . . . 15 ⊢ (𝐿 |s 𝑅) ∈ V
94	93	snid 4623	. . . . . . . . . . . . . 14 ⊢ (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}
95	28, 94	eqeltrdi 2872	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ {(𝐿 |s 𝑅)})
96	95	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
97	92, 39, 96	sltssepcd 28191	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 <s 𝐴)
98		cutcuts 28200	. . . . . . . . . . . . . . 15 ⊢ (𝑀 <<s 𝑆 → ((𝑀 |s 𝑆) ∈ No ∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
99	8, 98	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 |s 𝑆) ∈ No ∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
100	99	simp3d 1162	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(𝑀 |s 𝑆)} <<s 𝑆)
101	100	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → {(𝑀 |s 𝑆)} <<s 𝑆)
102		ovex 7479	. . . . . . . . . . . . . . 15 ⊢ (𝑀 |s 𝑆) ∈ V
103	102	snid 4623	. . . . . . . . . . . . . 14 ⊢ (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}
104	31, 103	eqeltrdi 2872	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ {(𝑀 |s 𝑆)})
105	104	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
106	101, 105, 47	sltssepcd 28191	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 <s 𝑢)
107	40, 43, 41, 48, 97, 106	ltmulsd 28556	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵)) <s ((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)))
108	34	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝐵) ∈ No)
109	51, 42, 49, 108	ltsubsubs2bd 28503	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵)) <s ((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) ↔ ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑢)) <s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢))))
110	42, 51	subscld 28482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢)) ∈ No)
111	108, 49, 110	ltsubaddsd 28508	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑢)) <s ((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢)) ↔ (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢))))
112	109, 111	bitrd 282	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝑡 ·s 𝑢) −s (𝑡 ·s 𝐵)) <s ((𝐴 ·s 𝑢) −s (𝐴 ·s 𝐵)) ↔ (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢))))
113	107, 112	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)))
114	42, 49, 51	addsubsd 28501	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) = (((𝑡 ·s 𝐵) −s (𝑡 ·s 𝑢)) +s (𝐴 ·s 𝑢)))
115	113, 114	breqtrrd 5137	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)))
116		breq2 5110	. . . . . . . 8 ⊢ (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) → ((𝐴 ·s 𝐵) <s 𝑦 ↔ (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))))
117	115, 116	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) → (𝐴 ·s 𝐵) <s 𝑦))
118	117	rexlimdvva 3223	. . . . . 6 ⊢ (𝜑 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) → (𝐴 ·s 𝐵) <s 𝑦))
119	90	simp3d 1162	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅)
120	119	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → {(𝐿 |s 𝑅)} <<s 𝑅)
121	95	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
122	120, 121, 60	sltssepcd 28191	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 <s 𝑣)
123	99	simp2d 1161	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 <<s {(𝑀 |s 𝑆)})
124	123	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑀 <<s {(𝑀 |s 𝑆)})
125	104	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
126	124, 68, 125	sltssepcd 28191	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 <s 𝐵)
127	64, 61, 69, 62, 122, 126	ltmulsd 28556	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤)) <s ((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)))
128	34	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝐵) ∈ No)
129	63, 72	subscld 28482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) ∈ No)
130	128, 70, 129	ltsubaddsd 28508	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝐴 ·s 𝐵) −s (𝐴 ·s 𝑤)) <s ((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) ↔ (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤))))
131	127, 130	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)))
132	63, 70, 72	addsubsd 28501	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) = (((𝑣 ·s 𝐵) −s (𝑣 ·s 𝑤)) +s (𝐴 ·s 𝑤)))
133	131, 132	breqtrrd 5137	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)))
134		breq2 5110	. . . . . . . 8 ⊢ (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) → ((𝐴 ·s 𝐵) <s 𝑦 ↔ (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))))
135	133, 134	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) → (𝐴 ·s 𝐵) <s 𝑦))
136	135	rexlimdvva 3223	. . . . . 6 ⊢ (𝜑 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤)) → (𝐴 ·s 𝐵) <s 𝑦))
137	118, 136	jaod 873	. . . . 5 ⊢ (𝜑 → ((∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢)) ∨ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))) → (𝐴 ·s 𝐵) <s 𝑦))
138	88, 137	biimtrid 245	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) → (𝐴 ·s 𝐵) <s 𝑦))
139		velsn 4600	. . . . 5 ⊢ (𝑥 ∈ {(𝐴 ·s 𝐵)} ↔ 𝑥 = (𝐴 ·s 𝐵))
140		breq1 5109	. . . . . 6 ⊢ (𝑥 = (𝐴 ·s 𝐵) → (𝑥 <s 𝑦 ↔ (𝐴 ·s 𝐵) <s 𝑦))
141	140	imbi2d 343	. . . . 5 ⊢ (𝑥 = (𝐴 ·s 𝐵) → ((𝑦 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) → 𝑥 <s 𝑦) ↔ (𝑦 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) → (𝐴 ·s 𝐵) <s 𝑦)))
142	139, 141	sylbi 220	. . . 4 ⊢ (𝑥 ∈ {(𝐴 ·s 𝐵)} → ((𝑦 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) → 𝑥 <s 𝑦) ↔ (𝑦 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) → (𝐴 ·s 𝐵) <s 𝑦)))
143	138, 142	syl5ibrcom 250	. . 3 ⊢ (𝜑 → (𝑥 ∈ {(𝐴 ·s 𝐵)} → (𝑦 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) → 𝑥 <s 𝑦)))
144	143	3imp 1128	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ {(𝐴 ·s 𝐵)} ∧ 𝑦 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))})) → 𝑥 <s 𝑦)
145	2, 27, 35, 78, 144	sltsd 28187	1 ⊢ (𝜑 → {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  {csn 4588   class class class wbr 5106  ran crn 5657  (class class class)co 7446   ∈ cmpo 7448  Nocsur 28030   <s clts 28031   <<s cslts 28176   |s ccuts 28178   +_s cadds 28378   −_s csubs 28439   ·_s cmuls 28525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440  df-subs 28441  df-muls 28526

This theorem is used by:  mulsuniflem  28568