Theorem ssltmul2 28051

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 5391	. . 3 ⊢ {(𝐴 ·s 𝐵)} ∈ V
2	1	a1i 11	. 2 ⊢ (𝜑 → {(𝐴 ·s 𝐵)} ∈ V)
3		eqid 2729	. . . . 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 7522	. . . 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 27698	. . . . . . 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 27699	. . . . . . 7 ⊢ (𝑀 <<s 𝑆 → 𝑆 ∈ V)
10	8, 9	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
11	3	mpoexg 8055	. . . . . 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 7878	. . . . 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 2833	. . 3 ⊢ (𝜑 → {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∈ V)
16		eqid 2729	. . . . 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 7522	. . . 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 27699	. . . . . . 7 ⊢ (𝐿 <<s 𝑅 → 𝑅 ∈ V)
19	5, 18	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ V)
20		ssltex1 27698	. . . . . . 7 ⊢ (𝑀 <<s 𝑆 → 𝑀 ∈ V)
21	8, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ V)
22	16	mpoexg 8055	. . . . . 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 7878	. . . . 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 2833	. . 3 ⊢ (𝜑 → {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ∈ V)
27	15, 26	unexd 7730	. 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 27715	. . . . 5 ⊢ (𝜑 → (𝐿 |s 𝑅) ∈ No )
30	28, 29	eqeltrd 2828	. . . 4 ⊢ (𝜑 → 𝐴 ∈ No )
31		ssltmul2.4	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))
32	8	scutcld 27715	. . . . 5 ⊢ (𝜑 → (𝑀 |s 𝑆) ∈ No )
33	31, 32	eqeltrd 2828	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No )
34	30, 33	mulscld 28038	. . 3 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
35	34	snssd 4773	. 2 ⊢ (𝜑 → {(𝐴 ·s 𝐵)} ⊆ No )
36		ssltss1 27700	. . . . . . . . . . . 12 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No )
37	5, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ⊆ No )
38	37	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐿 ⊆ No )
39		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ 𝐿)
40	38, 39	sseldd 3947	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ No )
41	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ No )
42	40, 41	mulscld 28038	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝐵) ∈ No )
43	30	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐴 ∈ No )
44		ssltss2 27701	. . . . . . . . . . . 12 ⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆ No )
45	8, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ No )
46	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ⊆ No )
47		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆)
48	46, 47	sseldd 3947	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ No )
49	43, 48	mulscld 28038	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝑢) ∈ No )
50	42, 49	addscld 27887	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No )
51	40, 48	mulscld 28038	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑡 ·s 𝑢) ∈ No )
52	50, 51	subscld 27967	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No )
53		eleq1 2816	. . . . . 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 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑐 ∈ No ))
55	54	rexlimdvva 3194	. . . 4 ⊢ (𝜑 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑐 ∈ No ))
56	55	abssdv 4031	. . 3 ⊢ (𝜑 → {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ⊆ No )
57		ssltss2 27701	. . . . . . . . . . . 12 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No )
58	5, 57	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ⊆ No )
59	58	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑅 ⊆ No )
60		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ 𝑅)
61	59, 60	sseldd 3947	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ No )
62	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ No )
63	61, 62	mulscld 28038	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝐵) ∈ No )
64	30	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ No )
65		ssltss1 27700	. . . . . . . . . . . 12 ⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆ No )
66	8, 65	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ⊆ No )
67	66	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑀 ⊆ No )
68		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ 𝑀)
69	67, 68	sseldd 3947	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ No )
70	64, 69	mulscld 28038	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝑤) ∈ No )
71	63, 70	addscld 27887	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) ∈ No )
72	61, 69	mulscld 28038	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·s 𝑤) ∈ No )
73	71, 72	subscld 27967	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No )
74		eleq1 2816	. . . . . 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 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑑 ∈ No ))
76	75	rexlimdvva 3194	. . . 4 ⊢ (𝜑 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑑 ∈ No ))
77	76	abssdv 4031	. . 3 ⊢ (𝜑 → {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ⊆ No )
78	56, 77	unssd 4155	. 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 4116	. . . . . 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 3451	. . . . . . . 8 ⊢ 𝑦 ∈ V
81		eqeq1 2733	. . . . . . . . 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 3202	. . . . . . . 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 3646	. . . . . . 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 2733	. . . . . . . . 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 3202	. . . . . . . 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 3646	. . . . . . 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 914	. . . . . 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 275	. . . . 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 27713	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐿 <<s {(𝐿 |s 𝑅)})
93		ovex 7420	. . . . . . . . . . . . . . 15 ⊢ (𝐿 |s 𝑅) ∈ V
94	93	snid 4626	. . . . . . . . . . . . . 14 ⊢ (𝐿 |s 𝑅) ∈ {(𝐿 |s 𝑅)}
95	28, 94	eqeltrdi 2836	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ {(𝐿 |s 𝑅)})
96	95	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
97	92, 39, 96	ssltsepcd 27706	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 <s 𝐴)
98		scutcut 27713	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → {(𝑀 |s 𝑆)} <<s 𝑆)
102		ovex 7420	. . . . . . . . . . . . . . 15 ⊢ (𝑀 |s 𝑆) ∈ V
103	102	snid 4626	. . . . . . . . . . . . . 14 ⊢ (𝑀 |s 𝑆) ∈ {(𝑀 |s 𝑆)}
104	31, 103	eqeltrdi 2836	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ {(𝑀 |s 𝑆)})
105	104	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
106	101, 105, 47	ssltsepcd 27706	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 <s 𝑢)
107	40, 43, 41, 48, 97, 106	sltmuld 28040	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝑢) -s (𝑡 ·s 𝐵)) <s ((𝐴 ·s 𝑢) -s (𝐴 ·s 𝐵)))
108	34	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝐵) ∈ No )
109	51, 42, 49, 108	sltsubsub2bd 27988	. . . . . . . . . . 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 27967	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑡 ·s 𝐵) -s (𝑡 ·s 𝑢)) ∈ No )
111	108, 49, 110	sltsubaddd 27993	. . . . . . . . . . 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 279	. . . . . . . . . 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 232	. . . . . . . . 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 27986	. . . . . . . . 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 5135	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → (𝐴 ·s 𝐵) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
116		breq2 5111	. . . . . . . 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 247	. . . . . . 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 3194	. . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → {(𝐿 |s 𝑅)} <<s 𝑅)
121	95	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ {(𝐿 |s 𝑅)})
122	120, 121, 60	ssltsepcd 27706	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 <s 𝑣)
123	99	simp2d 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 <<s {(𝑀 |s 𝑆)})
124	123	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑀 <<s {(𝑀 |s 𝑆)})
125	104	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ {(𝑀 |s 𝑆)})
126	124, 68, 125	ssltsepcd 27706	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 <s 𝐵)
127	64, 61, 69, 62, 122, 126	sltmuld 28040	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝑤)) <s ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)))
128	34	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝐵) ∈ No )
129	63, 72	subscld 27967	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ·s 𝐵) -s (𝑣 ·s 𝑤)) ∈ No )
130	128, 70, 129	sltsubaddd 27993	. . . . . . . . . 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 232	. . . . . . . . 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 27986	. . . . . . . . 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 5135	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·s 𝐵) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
134		breq2 5111	. . . . . . . 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 247	. . . . . . 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 3194	. . . . . 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 859	. . . . 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 242	. . . 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 4605	. . . . 5 ⊢ (𝑥 ∈ {(𝐴 ·s 𝐵)} ↔ 𝑥 = (𝐴 ·s 𝐵))
140		breq1 5110	. . . . . 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 217	. . . 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 247	. . 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 1110	. 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 27703	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3447   ∪ cun 3912   ⊆ wss 3914  {csn 4589   class class class wbr 5107  ran crn 5639  (class class class)co 7387   ∈ cmpo 7389   No csur 27551   <s cslt 27552   <<s csslt 27692   |s cscut 27694   +_s cadds 27866   -_s csubs 27926   ·_s cmuls 28009

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-2o 8435  df-nadd 8630  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec 27845  df-norec2 27856  df-adds 27867  df-negs 27927  df-subs 27928  df-muls 28010

This theorem is referenced by:  mulsuniflem  28052