Theorem gsumle 31252

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 3939	. . . 4 ⊢ 𝐴 ⊆ 𝐴
3		sseq1 3942	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4	3	anbi2d 628	. . . . . . 7 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
5		reseq2 5875	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝐹 ↾ 𝑎) = (𝐹 ↾ ∅))
6	5	oveq2d 7271	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑀 Σg (𝐹 ↾ 𝑎)) = (𝑀 Σg (𝐹 ↾ ∅)))
7		reseq2 5875	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝐺 ↾ 𝑎) = (𝐺 ↾ ∅))
8	7	oveq2d 7271	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑀 Σg (𝐺 ↾ 𝑎)) = (𝑀 Σg (𝐺 ↾ ∅)))
9	6, 8	breq12d 5083	. . . . . . 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 3942	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑎 ⊆ 𝐴 ↔ 𝑒 ⊆ 𝐴))
12	11	anbi2d 628	. . . . . . 7 ⊢ (𝑎 = 𝑒 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝑒 ⊆ 𝐴)))
13		reseq2 5875	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (𝐹 ↾ 𝑎) = (𝐹 ↾ 𝑒))
14	13	oveq2d 7271	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑀 Σg (𝐹 ↾ 𝑎)) = (𝑀 Σg (𝐹 ↾ 𝑒)))
15		reseq2 5875	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (𝐺 ↾ 𝑎) = (𝐺 ↾ 𝑒))
16	15	oveq2d 7271	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑀 Σg (𝐺 ↾ 𝑎)) = (𝑀 Σg (𝐺 ↾ 𝑒)))
17	14, 16	breq12d 5083	. . . . . . 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 3942	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝑎 ⊆ 𝐴 ↔ (𝑒 ∪ {𝑦}) ⊆ 𝐴))
20	19	anbi2d 628	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴)))
21		reseq2 5875	. . . . . . . . 9 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝐹 ↾ 𝑎) = (𝐹 ↾ (𝑒 ∪ {𝑦})))
22	21	oveq2d 7271	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝑀 Σg (𝐹 ↾ 𝑎)) = (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))))
23		reseq2 5875	. . . . . . . . 9 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝐺 ↾ 𝑎) = (𝐺 ↾ (𝑒 ∪ {𝑦})))
24	23	oveq2d 7271	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑦}) → (𝑀 Σg (𝐺 ↾ 𝑎)) = (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))))
25	22, 24	breq12d 5083	. . . . . . 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 3942	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
28	27	anbi2d 628	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐴)))
29		reseq2 5875	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝐹 ↾ 𝑎) = (𝐹 ↾ 𝐴))
30	29	oveq2d 7271	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑀 Σg (𝐹 ↾ 𝑎)) = (𝑀 Σg (𝐹 ↾ 𝐴)))
31		reseq2 5875	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝐺 ↾ 𝑎) = (𝐺 ↾ 𝐴))
32	31	oveq2d 7271	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑀 Σg (𝐺 ↾ 𝑎)) = (𝑀 Σg (𝐺 ↾ 𝐴)))
33	30, 32	breq12d 5083	. . . . . . 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 31233	. . . . . . . . . 10 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
37		tospos 18053	. . . . . . . . . 10 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset)
38	35, 36, 37	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ Poset)
39		res0 5884	. . . . . . . . . . . 12 ⊢ (𝐹 ↾ ∅) = ∅
40	39	oveq2i 7266	. . . . . . . . . . 11 ⊢ (𝑀 Σg (𝐹 ↾ ∅)) = (𝑀 Σg ∅)
41		eqid 2738	. . . . . . . . . . . 12 ⊢ (0g‘𝑀) = (0g‘𝑀)
42	41	gsum0 18283	. . . . . . . . . . 11 ⊢ (𝑀 Σg ∅) = (0g‘𝑀)
43	40, 42	eqtri 2766	. . . . . . . . . 10 ⊢ (𝑀 Σg (𝐹 ↾ ∅)) = (0g‘𝑀)
44		omndmnd 31232	. . . . . . . . . . 11 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
45		gsumle.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑀)
46	45, 41	mndidcl 18315	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ 𝐵)
47	35, 44, 46	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑀) ∈ 𝐵)
48	43, 47	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ ∅)) ∈ 𝐵)
49		gsumle.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝑀)
50	45, 49	posref 17951	. . . . . . . . 9 ⊢ ((𝑀 ∈ Poset ∧ (𝑀 Σg (𝐹 ↾ ∅)) ∈ 𝐵) → (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐹 ↾ ∅)))
51	38, 48, 50	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐹 ↾ ∅)))
52		res0 5884	. . . . . . . . . 10 ⊢ (𝐺 ↾ ∅) = ∅
53	39, 52	eqtr4i 2769	. . . . . . . . 9 ⊢ (𝐹 ↾ ∅) = (𝐺 ↾ ∅)
54	53	oveq2i 7266	. . . . . . . 8 ⊢ (𝑀 Σg (𝐹 ↾ ∅)) = (𝑀 Σg (𝐺 ↾ ∅))
55	51, 54	breqtrdi 5111	. . . . . . 7 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐺 ↾ ∅)))
56	55	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐴) → (𝑀 Σg (𝐹 ↾ ∅)) ≤ (𝑀 Σg (𝐺 ↾ ∅)))
57		ssun1 4102	. . . . . . . . . 10 ⊢ 𝑒 ⊆ (𝑒 ∪ {𝑦})
58		sstr2 3924	. . . . . . . . . 10 ⊢ (𝑒 ⊆ (𝑒 ∪ {𝑦}) → ((𝑒 ∪ {𝑦}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴))
59	57, 58	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑒 ∪ {𝑦}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴)
60	59	anim2i 616	. . . . . . . 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 765	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴)) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → (𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴))
63		simpllr 772	. . . . . . . . . 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 2738	. . . . . . . . . . . 12 ⊢ (+g‘𝑀) = (+g‘𝑀)
66	35	ad3antrrr 726	. . . . . . . . . . . 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 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐺:𝐴⟶𝐵)
69		simplr 765	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑒 ∪ {𝑦}) ⊆ 𝐴)
70		ssun2 4103	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦} ⊆ (𝑒 ∪ {𝑦})
71		vex 3426	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
72	71	snss 4716	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝑒 ∪ {𝑦}) ↔ {𝑦} ⊆ (𝑒 ∪ {𝑦}))
73	70, 72	mpbir 230	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ (𝑒 ∪ {𝑦})
74	73	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑦 ∈ (𝑒 ∪ {𝑦}))
75	69, 74	sseldd 3918	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑦 ∈ 𝐴)
76	68, 75	ffvelrnd 6944	. . . . . . . . . . . . 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 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑀 ∈ CMnd)
80		vex 3426	. . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐹:𝐴⟶𝐵)
84	57, 69	sstrid 3928	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑒 ⊆ 𝐴)
85	83, 84	fssresd 6625	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹 ↾ 𝑒):𝑒⟶𝐵)
86	1	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐴 ∈ Fin)
87		fvexd 6771	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (0g‘𝑀) ∈ V)
88	83, 86, 87	fdmfifsupp 9068	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐹 finSupp (0g‘𝑀))
89	88, 87	fsuppres 9083	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹 ↾ 𝑒) finSupp (0g‘𝑀))
90	45, 41, 79, 81, 85, 89	gsumcl 19431	. . . . . . . . . . . . 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	ffvelrnd 6944	. . . . . . . . . . . . 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 6625	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐺 ↾ 𝑒):𝑒⟶𝐵)
95		ssfi 8918	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Fin ∧ 𝑒 ⊆ 𝐴) → 𝑒 ∈ Fin)
96	86, 84, 95	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑒 ∈ Fin)
97	94, 96, 87	fdmfifsupp 9068	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐺 ↾ 𝑒) finSupp (0g‘𝑀))
98	45, 41, 79, 81, 94, 97	gsumcl 19431	. . . . . . . . . . . . 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 763	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝜑)
102		gsumle.c	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∘r ≤ 𝐺)
103	102	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝐹 ∘r ≤ 𝐺)
104	82	ffnd 6585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 Fn 𝐴)
105	67	ffnd 6585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 Fn 𝐴)
106		inidm 4149	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ 𝐴) = 𝐴
107		eqidd 2739	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
108		eqidd 2739	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = (𝐺‘𝑦))
109	104, 105, 1, 1, 106, 107, 108	ofrval 7523	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐹 ∘r ≤ 𝐺 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ (𝐺‘𝑦))
110	101, 103, 75, 109	syl3anc 1369	. . . . . . . . . . . . 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 31236	. . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 ∈ 𝑒) → 𝐹:𝐴⟶𝐵)
116		simplr 765	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 ∈ 𝑒) → (𝑒 ∪ {𝑦}) ⊆ 𝐴)
117		elun1 4106	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑒 → 𝑧 ∈ (𝑒 ∪ {𝑦}))
118	117	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 ∈ 𝑒) → 𝑧 ∈ (𝑒 ∪ {𝑦}))
119	116, 118	sseldd 3918	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ 𝑧 ∈ 𝑒) → 𝑧 ∈ 𝐴)
120	115, 119	ffvelrnd 6944	. . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . 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 765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ (𝑀 Σg (𝐹 ↾ 𝑒)) ≤ (𝑀 Σg (𝐺 ↾ 𝑒))) → ¬ 𝑦 ∈ 𝑒)
126		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
127	45, 65, 112, 114, 123, 124, 125, 93, 126	gsumunsn 19476	. . . . . . . . . . . 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 6820	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹 ↾ (𝑒 ∪ {𝑦})) = (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐹‘𝑧)))
129	128	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐹 ↾ (𝑒 ∪ {𝑦}))) = (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐹‘𝑧))))
130	83, 84	feqresmpt 6820	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐹 ↾ 𝑒) = (𝑧 ∈ 𝑒 ↦ (𝐹‘𝑧)))
131	130	oveq2d 7271	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐹 ↾ 𝑒)) = (𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐹‘𝑧))))
132	131	oveq1d 7270	. . . . . . . . . . . . . 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 2754	. . . . . . . . . . . . 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 256	. . . . . . . . . . 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 722	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ 𝑧 ∈ 𝑒) → 𝐺:𝐴⟶𝐵)
138	119	adantlr 711	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) ∧ 𝑧 ∈ 𝑒) → 𝑧 ∈ 𝐴)
139	137, 138	ffvelrnd 6944	. . . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → (𝐺‘𝑧) = (𝐺‘𝑦))
143	45, 65, 79, 96, 139, 140, 141, 76, 142	gsumunsn 19476	. . . . . . . . . . . . . . 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 6820	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝐺 ↾ (𝑒 ∪ {𝑦})) = (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐺‘𝑧)))
146	145	oveq2d 7271	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg (𝐺 ↾ (𝑒 ∪ {𝑦}))) = (𝑀 Σg (𝑧 ∈ (𝑒 ∪ {𝑦}) ↦ (𝐺‘𝑧))))
147		resabs1 5910	. . . . . . . . . . . . . . . . . . . . 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 6820	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝐺 ↾ 𝑒) = (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧)))
151	148, 150	eqtrd 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒) = (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧)))
152	151	oveq2d 7271	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ 𝑒)) = (𝑀 Σg (𝑧 ∈ 𝑒 ↦ (𝐺‘𝑧))))
153		resabs1 5910	. . . . . . . . . . . . . . . . . . . . . 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 3928	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → {𝑦} ⊆ 𝐴)
156	136, 155	feqresmpt 6820	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝐺 ↾ {𝑦}) = (𝑧 ∈ {𝑦} ↦ (𝐺‘𝑧)))
157	154, 156	eqtrd 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦}) = (𝑧 ∈ {𝑦} ↦ (𝐺‘𝑧)))
158	157	oveq2d 7271	. . . . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → 𝑦 ∈ 𝐴)
164	136, 163	ffvelrnd 6944	. . . . . . . . . . . . . . . . . . . 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 19468	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg (𝑧 ∈ {𝑦} ↦ (𝐺‘𝑧))) = (𝐺‘𝑦))
167	158, 166	eqtrd 2778	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) → (𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦})) = (𝐺‘𝑦))
168	152, 167	oveq12d 7273	. . . . . . . . . . . . . . . . 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 2754	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . 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 7266	. . . . . . . . . . . . . . 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 7266	. . . . . . . . . . . . . . 15 ⊢ (𝑀 Σg ((𝐺 ↾ (𝑒 ∪ {𝑦})) ↾ {𝑦})) = (𝑀 Σg (𝐺 ↾ {𝑦}))
176	173, 175	oveq12i 7267	. . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . 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 3928	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → {𝑦} ⊆ 𝐴)
179	68, 178	feqresmpt 6820	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝐺 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐺‘𝑥)))
180	179	oveq2d 7271	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐺 ↾ {𝑦})) = (𝑀 Σg (𝑥 ∈ {𝑦} ↦ (𝐺‘𝑥))))
181		cmnmnd 19317	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ CMnd → 𝑀 ∈ Mnd)
182	79, 181	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → 𝑀 ∈ Mnd)
183		fveq2 6756	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦))
184	45, 183	gsumsn 19470	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ Mnd ∧ 𝑦 ∈ V ∧ (𝐺‘𝑦) ∈ 𝐵) → (𝑀 Σg (𝑥 ∈ {𝑦} ↦ (𝐺‘𝑥))) = (𝐺‘𝑦))
185	182, 140, 76, 184	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝑥 ∈ {𝑦} ↦ (𝐺‘𝑥))) = (𝐺‘𝑦))
186	180, 185	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑒 ∪ {𝑦}) ⊆ 𝐴) ∧ ¬ 𝑦 ∈ 𝑒) → (𝑀 Σg (𝐺 ↾ {𝑦})) = (𝐺‘𝑦))
187	186	oveq2d 7271	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 5102	. . . . . . . . . 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 834	. . . . . . . . 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 8910	. . . . 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 700	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝜑) → (𝑀 Σg (𝐹 ↾ 𝐴)) ≤ (𝑀 Σg (𝐺 ↾ 𝐴)))
198	1, 197	mpancom 684	. 2 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ 𝐴)) ≤ (𝑀 Σg (𝐺 ↾ 𝐴)))
199		fnresdm 6535	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
200	104, 199	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹)
201	200	oveq2d 7271	. 2 ⊢ (𝜑 → (𝑀 Σg (𝐹 ↾ 𝐴)) = (𝑀 Σg 𝐹))
202		fnresdm 6535	. . . 4 ⊢ (𝐺 Fn 𝐴 → (𝐺 ↾ 𝐴) = 𝐺)
203	105, 202	syl 17	. . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐴) = 𝐺)
204	203	oveq2d 7271	. 2 ⊢ (𝜑 → (𝑀 Σg (𝐺 ↾ 𝐴)) = (𝑀 Σg 𝐺))
205	198, 201, 204	3brtr3d 5101	1 ⊢ (𝜑 → (𝑀 Σg 𝐹) ≤ (𝑀 Σg 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558   class class class wbr 5070   ↦ cmpt 5153   ↾ cres 5582   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_r cofr 7510  Fincfn 8691  Basecbs 16840  +_gcplusg 16888  lecple 16895  0_gc0g 17067   Σ_g cgsu 17068  Posetcpo 17940  Tosetctos 18049  Mndcmnd 18300  CMndccmn 19301  oMndcomnd 31225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-0g 17069  df-gsum 17070  df-mre 17212  df-mrc 17213  df-acs 17215  df-proset 17928  df-poset 17946  df-toset 18050  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-omnd 31227

This theorem is referenced by: (None)