Theorem gsumle 20074

Description: A finite sum in an ordered monoid is monotonic. This proof would be much easier in an ordered group, where an inverse element would be available. (Contributed by Thierry Arnoux, 13-Mar-2018.)

Hypotheses

Ref	Expression
gsumle.b	⊢ 𝐵 = (Base‘𝑀)
gsumle.l	⊢ ≤ = (le‘𝑀)
gsumle.m	⊢ (𝜑 → 𝑀 ∈ oMnd)
gsumle.n	⊢ (𝜑 → 𝑀 ∈ CMnd)
gsumle.a	⊢ (𝜑 → 𝐴 ∈ Fin)
gsumle.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumle.g	⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
gsumle.c	⊢ (𝜑 → 𝐹 ∘r ≤ 𝐺)

Assertion

Ref	Expression
gsumle	⊢ (𝜑 → (𝑀 Σg 𝐹) ≤ (𝑀 Σg 𝐺))

Proof of Theorem gsumle

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

Step	Hyp	Ref	Expression
1		gsumle.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		ssid 3956	. . . 4 ⊢ 𝐴 ⊆ 𝐴
3		sseq1 3959	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4	3	anbi2d 630	. . . . . . 7 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
5		reseq2 5933	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝐹 ↾ 𝑎) = (𝐹 ↾ ∅))
6	5	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑀 Σg (𝐹 ↾ 𝑎)) = (𝑀 Σg (𝐹 ↾ ∅)))
7		reseq2 5933	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝐺 ↾ 𝑎) = (𝐺 ↾ ∅))
8	7	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑀 Σg (𝐺 ↾ 𝑎)) = (𝑀 Σg (𝐺 ↾ ∅)))
9	6, 8	breq12d 5111	. . . . . . 7 ⊢ (𝑎 = ∅ → ((𝑀 Σg (𝐹 ↾ 𝑎)) ≤ (𝑀 Σg (𝐺 ↾ 𝑎)) ↔ (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐺 ↾ ∅))))
10	4, 9	imbi12d 344	. . . . . 6 ⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝑎)) ≤ (𝑀 Σg (𝐺 ↾ 𝑎))) ↔ ((𝜑 ∧ ∅ ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐺 ↾ ∅)))))
11		sseq1 3959	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑎 ⊆ 𝐴 ↔ 𝑒 ⊆ 𝐴))
12	11	anbi2d 630	. . . . . . 7 ⊢ (𝑎 = 𝑒 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝑒 ⊆ 𝐴)))
13		reseq2 5933	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (𝐹 ↾ 𝑎) = (𝐹 ↾ 𝑒))
14	13	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑀 Σg (𝐹 ↾ 𝑎)) = (𝑀 Σg (𝐹 ↾ 𝑒)))
15		reseq2 5933	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (𝐺 ↾ 𝑎) = (𝐺 ↾ 𝑒))
16	15	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑀 Σg (𝐺 ↾ 𝑎)) = (𝑀 Σg (𝐺 ↾ 𝑒)))
17	14, 16	breq12d 5111	. . . . . . 7 ⊢ (𝑎 = 𝑒 → ((𝑀 Σg (𝐹 ↾ 𝑎)) ≤ (𝑀 Σg (𝐺 ↾ 𝑎)) ↔ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))))
18	12, 17	imbi12d 344	. . . . . 6 ⊢ (𝑎 = 𝑒 → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝑎)) ≤ (𝑀 Σg (𝐺 ↾ 𝑎))) ↔ ((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒)))))
19		sseq1 3959	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝑎 ⊆ 𝐴 ↔ (𝑒 ∪ {𝑦}) ⊆ 𝐴))
20	19	anbi2d 630	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴)))
21		reseq2 5933	. . . . . . . . 9 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝐹 ↾ 𝑎) = (𝐹 ↾ (𝑒 ∪ {𝑦})))
22	21	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝑀 Σg (𝐹 ↾ 𝑎)) = (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))))
23		reseq2 5933	. . . . . . . . 9 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝐺 ↾ 𝑎) = (𝐺 ↾ (𝑒 ∪ {𝑦})))
24	23	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝑀 Σg (𝐺 ↾ 𝑎)) = (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))))
25	22, 24	breq12d 5111	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → ((𝑀 Σg (𝐹 ↾ 𝑎)) ≤ (𝑀 Σg (𝐺 ↾ 𝑎)) ↔ (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) ≤ (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦})))))
26	20, 25	imbi12d 344	. . . . . 6 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝑎)) ≤ (𝑀 Σg (𝐺 ↾ 𝑎))) ↔ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) ≤ (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))))))
27		sseq1 3959	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
28	27	anbi2d 630	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐴)))
29		reseq2 5933	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝐹 ↾ 𝑎) = (𝐹 ↾ 𝐴))
30	29	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑀 Σg (𝐹 ↾ 𝑎)) = (𝑀 Σg (𝐹 ↾ 𝐴)))
31		reseq2 5933	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝐺 ↾ 𝑎) = (𝐺 ↾ 𝐴))
32	31	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑀 Σg (𝐺 ↾ 𝑎)) = (𝑀 Σg (𝐺 ↾ 𝐴)))
33	30, 32	breq12d 5111	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑀 Σg (𝐹 ↾ 𝑎)) ≤ (𝑀 Σg (𝐺 ↾ 𝑎)) ↔ (𝑀 Σg (𝐹 ↾ 𝐴)) ≤ (𝑀 Σg (𝐺 ↾ 𝐴))))
34	28, 33	imbi12d 344	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝑎)) ≤ (𝑀 Σg (𝐺 ↾ 𝑎))) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝐴)) ≤ (𝑀 Σg (𝐺 ↾ 𝐴)))))
35		gsumle.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ oMnd)
36		omndtos 20056	. . . . . . . . . 10 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
37		tospos 18341	. . . . . . . . . 10 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset)
38	35, 36, 37	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ Poset)
39		res0 5942	. . . . . . . . . . . 12 ⊢ (𝐹 ↾ ∅) = ∅
40	39	oveq2i 7369	. . . . . . . . . . 11 ⊢ (𝑀 Σg (𝐹 ↾ ∅)) = (𝑀 Σg ∅)
41		eqid 2736	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
42	41	gsum0 18609	. . . . . . . . . . 11 ⊢ (𝑀 Σg ∅) = (0g‘𝑀)
43	40, 42	eqtri 2759	. . . . . . . . . 10 ⊢ (𝑀 Σg (𝐹 ↾ ∅)) = (0g‘𝑀)
44		omndmnd 20055	. . . . . . . . . . 11 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
45		gsumle.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑀)
46	45, 41	mndidcl 18674	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
47	35, 44, 46	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝐵)
48	43, 47	eqeltrid 2840	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ ∅)) ∈ 𝐵)
49		gsumle.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝑀)
50	45, 49	posref 18241	. . . . . . . . 9 ⊢ ((𝑀 ∈ Poset ∧ (𝑀 Σg (𝐹 ↾ ∅)) ∈ 𝐵) → (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐹 ↾ ∅)))
51	38, 48, 50	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐹 ↾ ∅)))
52		res0 5942	. . . . . . . . . 10 ⊢ (𝐺 ↾ ∅) = ∅
53	39, 52	eqtr4i 2762	. . . . . . . . 9 ⊢ (𝐹 ↾ ∅) = (𝐺 ↾ ∅)
54	53	oveq2i 7369	. . . . . . . 8 ⊢ (𝑀 Σg (𝐹 ↾ ∅)) = (𝑀 Σg (𝐺 ↾ ∅))
55	51, 54	breqtrdi 5139	. . . . . . 7 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐺 ↾ ∅)))
56	55	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐺 ↾ ∅)))
57		ssun1 4130	. . . . . . . . . 10 ⊢ 𝑒 ⊆ (𝑒 ∪ {𝑦})
58		sstr2 3940	. . . . . . . . . 10 ⊢ (𝑒 ⊆ (𝑒 ∪ {𝑦}) → ((𝑒 ∪ {𝑦}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴))
59	57, 58	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑒 ∪ {𝑦}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴)
60	59	anim2i 617	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝜑 ∧ 𝑒 ⊆ 𝐴))
61	60	imim1i 63	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))))
62		simplr 768	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴)) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴))
63		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴)) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → ¬ 𝑦 ∈ 𝑒)
64		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴)) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒)))
65		eqid 2736	. . . . . . . . . . . 12 ⊢ (+g‘𝑀) = (+g‘𝑀)
66	35	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → 𝑀 ∈ oMnd)
67		gsumle.g	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
68	67	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐺:𝐴⟶𝐵)
69		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑒 ∪ {𝑦}) ⊆ 𝐴)
70		ssun2 4131	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦} ⊆ (𝑒 ∪ {𝑦})
71		vex 3444	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
72	71	snss 4741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝑒 ∪ {𝑦}) ↔ {𝑦} ⊆ (𝑒 ∪ {𝑦}))
73	70, 72	mpbir 231	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ (𝑒 ∪ {𝑦})
74	73	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑦 ∈ (𝑒 ∪ {𝑦}))
75	69, 74	sseldd 3934	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑦 ∈ 𝐴)
76	68, 75	ffvelcdmd 7030	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐺‘𝑦) ∈ 𝐵)
77	76	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝐺‘𝑦) ∈ 𝐵)
78		gsumle.n	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ CMnd)
79	78	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑀 ∈ CMnd)
80		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
81	80	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑒 ∈ V)
82		gsumle.f	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
83	82	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐹:𝐴⟶𝐵)
84	57, 69	sstrid 3945	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑒 ⊆ 𝐴)
85	83, 84	fssresd 6701	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹 ↾ 𝑒):𝑒⟶𝐵)
86	1	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐴 ∈ Fin)
87		fvexd 6849	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (0g‘𝑀) ∈ V)
88	83, 86, 87	fdmfifsupp 9278	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐹 finSupp (0g‘𝑀))
89	88, 87	fsuppres 9296	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹 ↾ 𝑒) finSupp (0g‘𝑀))
90	45, 41, 79, 81, 85, 89	gsumcl 19844	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐹 ↾ 𝑒)) ∈ 𝐵)
91	90	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑀 Σg (𝐹 ↾ 𝑒)) ∈ 𝐵)
92	83, 75	ffvelcdmd 7030	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹‘𝑦) ∈ 𝐵)
93	92	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝐹‘𝑦) ∈ 𝐵)
94	68, 84	fssresd 6701	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐺 ↾ 𝑒):𝑒⟶𝐵)
95		ssfi 9097	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Fin ∧ 𝑒 ⊆ 𝐴) → 𝑒 ∈ Fin)
96	86, 84, 95	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑒 ∈ Fin)
97	94, 96, 87	fdmfifsupp 9278	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐺 ↾ 𝑒) finSupp (0g‘𝑀))
98	45, 41, 79, 81, 94, 97	gsumcl 19844	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐺 ↾ 𝑒)) ∈ 𝐵)
99	98	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑀 Σg (𝐺 ↾ 𝑒)) ∈ 𝐵)
100		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒)))
101		simpll 766	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝜑)
102		gsumle.c	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∘r ≤ 𝐺)
103	102	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐹 ∘r ≤ 𝐺)
104	82	ffnd 6663	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐴)
105	67	ffnd 6663	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 Fn 𝐴)
106		inidm 4179	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ 𝐴) = 𝐴
107		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
108		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
109	104, 105, 1, 1, 106, 107, 108	ofrval 7634	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∘r ≤ 𝐺 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ (𝐺‘𝑦))
110	101, 103, 75, 109	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹‘𝑦) ≤ (𝐺‘𝑦))
111	110	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝐹‘𝑦) ≤ (𝐺‘𝑦))
112	79	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → 𝑀 ∈ CMnd)
113	45, 49, 65, 66, 77, 91, 93, 99, 100, 111, 112	omndadd2d 20059	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → ((𝑀 Σg (𝐹 ↾ 𝑒))(+g‘𝑀)(𝐹‘𝑦)) ≤ ((𝑀 Σg (𝐺 ↾ 𝑒))(+g‘𝑀)(𝐺‘𝑦)))
114	96	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → 𝑒 ∈ Fin)
115	82	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 ∈ 𝑒) → 𝐹:𝐴⟶𝐵)
116		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 ∈ 𝑒) → (𝑒 ∪ {𝑦}) ⊆ 𝐴)
117		elun1 4134	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑒 → 𝑧 ∈ (𝑒 ∪ {𝑦}))
118	117	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 ∈ 𝑒) → 𝑧 ∈ (𝑒 ∪ {𝑦}))
119	116, 118	sseldd 3934	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 ∈ 𝑒) → 𝑧 ∈ 𝐴)
120	115, 119	ffvelcdmd 7030	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 ∈ 𝑒) → (𝐹‘𝑧) ∈ 𝐵)
121	120	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑧 ∈ 𝑒 → (𝐹‘𝑧) ∈ 𝐵))
122	121	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑧 ∈ 𝑒 → (𝐹‘𝑧) ∈ 𝐵))
123	122	imp 406	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) ∧ 𝑧 ∈ 𝑒) → (𝐹‘𝑧) ∈ 𝐵)
124	71	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → 𝑦 ∈ V)
125		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → ¬ 𝑦 ∈ 𝑒)
126		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
127	45, 65, 112, 114, 123, 124, 125, 93, 126	gsumunsn 19889	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐹‘𝑧))) = ((𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐹‘𝑧)))(+g‘𝑀)(𝐹‘𝑦)))
128	83, 69	feqresmpt 6903	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹 ↾ (𝑒 ∪ {𝑦})) = (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐹‘𝑧)))
129	128	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) = (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐹‘𝑧))))
130	83, 84	feqresmpt 6903	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹 ↾ 𝑒) = (𝑧 ∈ 𝑒 ↦ (𝐹‘𝑧)))
131	130	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐹 ↾ 𝑒)) = (𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐹‘𝑧))))
132	131	oveq1d 7373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → ((𝑀 Σg (𝐹 ↾ 𝑒))(+g‘𝑀)(𝐹‘𝑦)) = ((𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐹‘𝑧)))(+g‘𝑀)(𝐹‘𝑦)))
133	129, 132	eqeq12d 2752	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → ((𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) = ((𝑀 Σg (𝐹 ↾ 𝑒))(+g‘𝑀)(𝐹‘𝑦)) ↔ (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐹‘𝑧))) = ((𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐹‘𝑧)))(+g‘𝑀)(𝐹‘𝑦))))
134	133	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → ((𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) = ((𝑀 Σg (𝐹 ↾ 𝑒))(+g‘𝑀)(𝐹‘𝑦)) ↔ (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐹‘𝑧))) = ((𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐹‘𝑧)))(+g‘𝑀)(𝐹‘𝑦))))
135	127, 134	mpbird 257	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) = ((𝑀 Σg (𝐹 ↾ 𝑒))(+g‘𝑀)(𝐹‘𝑦)))
136	67	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → 𝐺:𝐴⟶𝐵)
137	136	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ 𝑧 ∈ 𝑒) → 𝐺:𝐴⟶𝐵)
138	119	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ 𝑧 ∈ 𝑒) → 𝑧 ∈ 𝐴)
139	137, 138	ffvelcdmd 7030	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ 𝑧 ∈ 𝑒) → (𝐺‘𝑧) ∈ 𝐵)
140	71	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑦 ∈ V)
141		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → ¬ 𝑦 ∈ 𝑒)
142		fveq2 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → (𝐺‘𝑧) = (𝐺‘𝑦))
143	45, 65, 79, 96, 139, 140, 141, 76, 142	gsumunsn 19889	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐺‘𝑧))) = ((𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧)))(+g‘𝑀)(𝐺‘𝑦)))
144		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑒 ∪ {𝑦}) ⊆ 𝐴)
145	136, 144	feqresmpt 6903	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝐺 ↾ (𝑒 ∪ {𝑦})) = (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐺‘𝑧)))
146	145	oveq2d 7374	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))) = (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐺‘𝑧))))
147		resabs1 5965	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 ⊆ (𝑒 ∪ {𝑦}) → ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒) = (𝐺 ↾ 𝑒))
148	57, 147	mp1i 13	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒) = (𝐺 ↾ 𝑒))
149	59	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → 𝑒 ⊆ 𝐴)
150	136, 149	feqresmpt 6903	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝐺 ↾ 𝑒) = (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧)))
151	148, 150	eqtrd 2771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒) = (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧)))
152	151	oveq2d 7374	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒)) = (𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧))))
153		resabs1 5965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({𝑦} ⊆ (𝑒 ∪ {𝑦}) → ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}) = (𝐺 ↾ {𝑦}))
154	70, 153	mp1i 13	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}) = (𝐺 ↾ {𝑦}))
155	70, 144	sstrid 3945	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → {𝑦} ⊆ 𝐴)
156	136, 155	feqresmpt 6903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝐺 ↾ {𝑦}) = (𝑧 ∈ {𝑦} ↦ (𝐺‘𝑧)))
157	154, 156	eqtrd 2771	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}) = (𝑧 ∈ {𝑦} ↦ (𝐺‘𝑧)))
158	157	oveq2d 7374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦})) = (𝑀 Σg (𝑧 ∈ {𝑦} ↦ (𝐺‘𝑧))))
159	35, 44	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ∈ Mnd)
160	159	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → 𝑀 ∈ Mnd)
161	71	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ V)
162	73	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ (𝑒 ∪ {𝑦}))
163	144, 162	sseldd 3934	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ 𝐴)
164	136, 163	ffvelcdmd 7030	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝐺‘𝑦) ∈ 𝐵)
165	142	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 = 𝑦) → (𝐺‘𝑧) = (𝐺‘𝑦))
166	45, 160, 161, 164, 165	gsumsnd 19881	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg (𝑧 ∈ {𝑦} ↦ (𝐺‘𝑧))) = (𝐺‘𝑦))
167	158, 166	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦})) = (𝐺‘𝑦))
168	152, 167	oveq12d 7376	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → ((𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒))(+g‘𝑀)(𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}))) = ((𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧)))(+g‘𝑀)(𝐺‘𝑦)))
169	146, 168	eqeq12d 2752	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → ((𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))) = ((𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒))(+g‘𝑀)(𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}))) ↔ (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐺‘𝑧))) = ((𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧)))(+g‘𝑀)(𝐺‘𝑦))))
170	169	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → ((𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))) = ((𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒))(+g‘𝑀)(𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}))) ↔ (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐺‘𝑧))) = ((𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧)))(+g‘𝑀)(𝐺‘𝑦))))
171	143, 170	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))) = ((𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒))(+g‘𝑀)(𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}))))
172	57, 147	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒) = (𝐺 ↾ 𝑒)
173	172	oveq2i 7369	. . . . . . . . . . . . . . 15 ⊢ (𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒)) = (𝑀 Σg (𝐺 ↾ 𝑒))
174	70, 153	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}) = (𝐺 ↾ {𝑦})
175	174	oveq2i 7369	. . . . . . . . . . . . . . 15 ⊢ (𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦})) = (𝑀 Σg (𝐺 ↾ {𝑦}))
176	173, 175	oveq12i 7370	. . . . . . . . . . . . . 14 ⊢ ((𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒))(+g‘𝑀)(𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}))) = ((𝑀 Σg (𝐺 ↾ 𝑒))(+g‘𝑀)(𝑀 Σg (𝐺 ↾ {𝑦})))
177	171, 176	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))) = ((𝑀 Σg (𝐺 ↾ 𝑒))(+g‘𝑀)(𝑀 Σg (𝐺 ↾ {𝑦}))))
178	70, 69	sstrid 3945	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → {𝑦} ⊆ 𝐴)
179	68, 178	feqresmpt 6903	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐺 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐺‘𝑥)))
180	179	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐺 ↾ {𝑦})) = (𝑀 Σg (𝑥 ∈ {𝑦} ↦ (𝐺‘𝑥))))
181		cmnmnd 19726	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ CMnd → 𝑀 ∈ Mnd)
182	79, 181	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑀 ∈ Mnd)
183		fveq2 6834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦))
184	45, 183	gsumsn 19883	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ Mnd ∧ 𝑦 ∈ V ∧ (𝐺‘𝑦) ∈ 𝐵) → (𝑀 Σg (𝑥 ∈ {𝑦} ↦ (𝐺‘𝑥))) = (𝐺‘𝑦))
185	182, 140, 76, 184	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝑥 ∈ {𝑦} ↦ (𝐺‘𝑥))) = (𝐺‘𝑦))
186	180, 185	eqtrd 2771	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐺 ↾ {𝑦})) = (𝐺‘𝑦))
187	186	oveq2d 7374	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → ((𝑀 Σg (𝐺 ↾ 𝑒))(+g‘𝑀)(𝑀 Σg (𝐺 ↾ {𝑦}))) = ((𝑀 Σg (𝐺 ↾ 𝑒))(+g‘𝑀)(𝐺‘𝑦)))
188	177, 187	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))) = ((𝑀 Σg (𝐺 ↾ 𝑒))(+g‘𝑀)(𝐺‘𝑦)))
189	188	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))) = ((𝑀 Σg (𝐺 ↾ 𝑒))(+g‘𝑀)(𝐺‘𝑦)))
190	113, 135, 189	3brtr4d 5130	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) ≤ (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))))
191	62, 63, 64, 190	syl21anc 837	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴)) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) ≤ (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))))
192	191	exp31 419	. . . . . . . 8 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑒) → ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → ((𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒)) → (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) ≤ (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))))))
193	192	a2d 29	. . . . . . 7 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑒) → (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) ≤ (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))))))
194	61, 193	syl5 34	. . . . . 6 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑒) → (((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) ≤ (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))))))
195	10, 18, 26, 34, 56, 194	findcard2s 9090	. . . . 5 ⊢ (𝐴 ∈ Fin → ((𝜑 ∧ 𝐴 ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ 𝐴)) ≤ (𝑀 Σg (𝐺 ↾ 𝐴))))
196	195	imp 406	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝜑 ∧ 𝐴 ⊆ 𝐴)) → (𝑀 Σg (𝐹 ↾ 𝐴)) ≤ (𝑀 Σg (𝐺 ↾ 𝐴)))
197	2, 196	mpanr2 704	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝜑) → (𝑀 Σg (𝐹 ↾ 𝐴)) ≤ (𝑀 Σg (𝐺 ↾ 𝐴)))
198	1, 197	mpancom 688	. 2 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ 𝐴)) ≤ (𝑀 Σg (𝐺 ↾ 𝐴)))
199		fnresdm 6611	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
200	104, 199	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹)
201	200	oveq2d 7374	. 2 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ 𝐴)) = (𝑀 Σg 𝐹))
202		fnresdm 6611	. . . 4 ⊢ (𝐺 Fn 𝐴 → (𝐺 ↾ 𝐴) = 𝐺)
203	105, 202	syl 17	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐴) = 𝐺)
204	203	oveq2d 7374	. 2 ⊢ (𝜑 → (𝑀 Σg (𝐺 ↾ 𝐴)) = (𝑀 Σg 𝐺))
205	198, 201, 204	3brtr3d 5129	1 ⊢ (𝜑 → (𝑀 Σg 𝐹) ≤ (𝑀 Σg 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  {csn 4580   class class class wbr 5098   ↦ cmpt 5179   ↾ cres 5626   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∘_r cofr 7621  Fincfn 8883  Basecbs 17136  +_gcplusg 17177  lecple 17184  0_gc0g 17359   Σ_g cgsu 17360  Posetcpo 18230  Tosetctos 18337  Mndcmnd 18659  CMndccmn 19709  oMndcomnd 20048

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-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

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-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-ofr 7623  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-0g 17361  df-gsum 17362  df-mre 17505  df-mrc 17506  df-acs 17508  df-proset 18217  df-poset 18236  df-toset 18338  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-mulg 18998  df-cntz 19246  df-cmn 19711  df-omnd 20050

This theorem is referenced by: (None)