Theorem ssltmul2 27532

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

Hypotheses

Ref	Expression
ssltmul2.1	⊢ (𝜑 → 𝐿 <<s 𝑅)
ssltmul2.2	⊢ (𝜑 → 𝑀 <<s 𝑆)
ssltmul2.3	⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
ssltmul2.4	⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))

Assertion

Ref	Expression
ssltmul2	⊢ (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 ssltmul2

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

Step	Hyp	Ref	Expression
1		snex 5425	. . 3 ⊢ {(𝐴 ·s 𝐵)} ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → {(𝐴 ·s 𝐵)} ∈ V)
3		eqid 2732	. . . . 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 7526	. . . 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		ssltmul2.1	. . . . . . 7 ⊢ (𝜑 → 𝐿 <<s 𝑅)
6		ssltex1 27217	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝐿 ∈ V)
7	5, 6	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ V)
8		ssltmul2.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 <<s 𝑆)
9		ssltex2 27218	. . . . . . 7 ⊢ (𝑀 <<s 𝑆 → 𝑆 ∈ V)
10	8, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
11	3	mpoexg 8047	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V)
13		rnexg 7879	. . . . 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 17	. . . 4 ⊢ (𝜑 → ran (𝑡 ∈ 𝐿, 𝑢 ∈ 𝑆 ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V)
15	4, 14	eqeltrrid 2838	. . 3 ⊢ (𝜑 → {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∈ V)
16		eqid 2732	. . . . 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 7526	. . . 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		ssltex2 27218	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝑅 ∈ V)
19	5, 18	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
20		ssltex1 27217	. . . . . . 7 ⊢ (𝑀 <<s 𝑆 → 𝑀 ∈ V)
21	8, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ V)
22	16	mpoexg 8047	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V)
24		rnexg 7879	. . . . 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 17	. . . 4 ⊢ (𝜑 → ran (𝑣 ∈ 𝑅, 𝑤 ∈ 𝑀 ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V)
26	17, 25	eqeltrrid 2838	. . 3 ⊢ (𝜑 → {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ∈ V)
27	15, 26	unexd 7725	. 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		ssltmul2.3	. . . . 5 ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
29	5	scutcld 27233	. . . . 5 ⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No )
30	28, 29	eqeltrd 2833	. . . 4 ⊢ (𝜑 → 𝐴 ∈ No )
31		ssltmul2.4	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))
32	8	scutcld 27233	. . . . 5 ⊢ (𝜑 → (𝑀 |s 𝑆) ∈ No )
33	31, 32	eqeltrd 2833	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No )
34	30, 33	mulscld 27520	. . 3 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
35	34	snssd 4806	. 2 ⊢ (𝜑 → {(𝐴 ·s 𝐵)} ⊆ No )
36		ssltss1 27219	. . . . . . . . . . . 12 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No )
37	5, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ⊆ No )
38	37	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐿 ⊆ No )
39		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ 𝐿)
40	38, 39	sseldd 3980	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ No )
41	33	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ No )
42	40, 41	mulscld 27520	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝐵) ∈ No )
43	30	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐴 ∈ No )
44		ssltss2 27220	. . . . . . . . . . . 12 ⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆ No )
45	8, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ No )
46	45	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ⊆ No )
47		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆)
48	46, 47	sseldd 3980	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ No )
49	43, 48	mulscld 27520	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝑢) ∈ No )
50	42, 49	addscld 27393	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No )
51	40, 48	mulscld 27520	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝑢) ∈ No )
52	50, 51	subscld 27464	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No )
53		eleq1 2821	. . . . . 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 246	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑐 ∈ No ))
55	54	rexlimdvva 3211	. . . 4 ⊢ (𝜑 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑐 ∈ No ))
56	55	abssdv 4062	. . 3 ⊢ (𝜑 → {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ⊆ No )
57		ssltss2 27220	. . . . . . . . . . . 12 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No )
58	5, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ⊆ No )
59	58	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑅 ⊆ No )
60		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ 𝑅)
61	59, 60	sseldd 3980	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ No )
62	33	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ No )
63	61, 62	mulscld 27520	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝐵) ∈ No )
64	30	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ No )
65		ssltss1 27219	. . . . . . . . . . . 12 ⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆ No )
66	8, 65	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ⊆ No )
67	66	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑀 ⊆ No )
68		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ 𝑀)
69	67, 68	sseldd 3980	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ No )
70	64, 69	mulscld 27520	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝑤) ∈ No )
71	63, 70	addscld 27393	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) ∈ No )
72	61, 69	mulscld 27520	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝑤) ∈ No )
73	71, 72	subscld 27464	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No )
74		eleq1 2821	. . . . . 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 246	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑑 ∈ No ))
76	75	rexlimdvva 3211	. . . 4 ⊢ (𝜑 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑑 ∈ No ))
77	76	abssdv 4062	. . 3 ⊢ (𝜑 → {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ⊆ No )
78	56, 77	unssd 4183	. 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 4145	. . . . . 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 3478	. . . . . . . 8 ⊢ 𝑦 ∈ V
81		eqeq1 2736	. . . . . . . . 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 3219	. . . . . . . 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 3665	. . . . . . 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 2736	. . . . . . . . 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 3219	. . . . . . . 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 3665	. . . . . . 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 913	. . . . . 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 274	. . . . 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		scutcut 27231	. . . . . . . . . . . . . . 15 ⊢ (𝐿 <<s 𝑅 → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
90	5, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐿 |s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 |s 𝑅)} ∧ {(𝐿 |s 𝑅)} <<s 𝑅))
91	90	simp2d 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 <<s {(𝐿 |s 𝑅)})
92	91	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐿 <<s {(𝐿 |s 𝑅)})
93		ovex 7427	. . . . . . . . . . . . . . 15 ⊢ (𝐿 |s 𝑅) ∈ V
94	93	snid 4659	. . . . . . . . . . . . . 14 ⊢ (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}
95	28, 94	eqeltrdi 2841	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ {(𝐿 |s 𝑅)})
96	95	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
97	92, 39, 96	ssltsepcd 27224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 <s 𝐴)
98		scutcut 27231	. . . . . . . . . . . . . . 15 ⊢ (𝑀 <<s 𝑆 → ((𝑀 |s 𝑆) ∈ No ∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
99	8, 98	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 |s 𝑆) ∈ No ∧ 𝑀 <<s {(𝑀 |s 𝑆)} ∧ {(𝑀 |s 𝑆)} <<s 𝑆))
100	99	simp3d 1144	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(𝑀 |s 𝑆)} <<s 𝑆)
101	100	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → {(𝑀 |s 𝑆)} <<s 𝑆)
102		ovex 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑀 |s 𝑆) ∈ V
103	102	snid 4659	. . . . . . . . . . . . . 14 ⊢ (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}
104	31, 103	eqeltrdi 2841	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ {(𝑀 |s 𝑆)})
105	104	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
106	101, 105, 47	ssltsepcd 27224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 <s 𝑢)
107	40, 43, 41, 48, 97, 106	sltmuld 27522	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)) <s ((𝐴 ·s 𝑢) -s (𝐴 ·s 𝐵)))
108	34	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝐵) ∈ No )
109	51, 42, 49, 108	sltsubsub2bd 27480	. . . . . . . . . . 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 27464	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) ∈ No )
111	108, 49, 110	sltsubaddd 27485	. . . . . . . . . . 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 278	. . . . . . . . . 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 231	. . . . . . . . 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 27478	. . . . . . . . 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 5170	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
116		breq2 5146	. . . . . . . 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 246	. . . . . . 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 3211	. . . . . 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 1144	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(𝐿 |s 𝑅)} <<s 𝑅)
120	119	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → {(𝐿 |s 𝑅)} <<s 𝑅)
121	95	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
122	120, 121, 60	ssltsepcd 27224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 <s 𝑣)
123	99	simp2d 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 <<s {(𝑀 |s 𝑆)})
124	123	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑀 <<s {(𝑀 |s 𝑆)})
125	104	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
126	124, 68, 125	ssltsepcd 27224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 <s 𝐵)
127	64, 61, 69, 62, 122, 126	sltmuld 27522	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑤)) <s ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)))
128	34	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝐵) ∈ No )
129	63, 72	subscld 27464	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)) ∈ No )
130	128, 70, 129	sltsubaddd 27485	. . . . . . . . . 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 231	. . . . . . . . 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 27478	. . . . . . . . 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 5170	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
134		breq2 5146	. . . . . . . 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 246	. . . . . . 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 3211	. . . . . 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 857	. . . . 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 241	. . . 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 4639	. . . . 5 ⊢ (𝑥 ∈ {(𝐴 ·s 𝐵)} ↔ 𝑥 = (𝐴 ·s 𝐵))
140		breq1 5145	. . . . . 6 ⊢ (𝑥 = (𝐴 ·s 𝐵) → (𝑥 <s 𝑦 ↔ (𝐴 ·s 𝐵) <s 𝑦))
141	140	imbi2d 340	. . . . 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 216	. . . 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 246	. . 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 1111	. 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	ssltd 27222	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))}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  ∃wrex 3070  Vcvv 3474   ∪ cun 3943   ⊆ wss 3945  {csn 4623   class class class wbr 5142  ran crn 5671  (class class class)co 7394   ∈ cmpo 7396   No csur 27072   <s cslt 27073   <<s csslt 27211   |s cscut 27213   +_s cadds 27372   -_s csubs 27424   ·_s cmuls 27491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-1o 8450  df-2o 8451  df-nadd 8650  df-no 27075  df-slt 27076  df-bday 27077  df-sle 27177  df-sslt 27212  df-scut 27214  df-0s 27254  df-made 27271  df-old 27272  df-left 27274  df-right 27275  df-norec 27351  df-norec2 27362  df-adds 27373  df-negs 27425  df-subs 27426  df-muls 27492

This theorem is referenced by:  mulsuniflem  27533